------------------------------ A Book of Definite Summation Fibonacci Identities in the style of Curtis Greene and Herbert Wilf By Shalosh B. Ekhad ------------------------------------------------ Theorem Number, 1 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = -1/6 F(n) (-1 - 3 F(n + 1) + 3 F(n) F(n + 1) + F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 2 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(2 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 G(n) = 5/22 - 5/22 F(n + 1) + 7/22 F(n) - 2/11 F(n) F(n + 1) 23 2 2 3 4 + -- F(n) F(n + 1) - 7/11 F(n) F(n + 1) - 6/11 F(n) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 3 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(2 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) G(n) = 1/3 F(n) F(n + 1) (F(n) - F(n + 1)) (F(n) - 2 F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 4 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(3 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 13 G(n) = 5/22 F(n + 1) - 5/22 F(n + 1) - -- F(n) + 8/11 F(n) F(n + 1) 22 2 31 2 3 3 2 - 3/11 F(n) + -- F(n) F(n + 1) - 5/2 F(n) F(n + 1) 22 4 19 5 + 4/11 F(n) F(n + 1) + -- F(n) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 5 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(3 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 2 2 G(n) = -1/6 F(n) (-5 + F(n + 1) - F(n) F(n + 1) - F(n) - 16 F(n) F(n + 1) 3 4 + 16 F(n) F(n + 1) + 6 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 6 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 105 105 2 21 2 21 5 G(n) = ---- F(n + 1) + --- F(n + 1) - --- F(n) + --- F(n) 638 638 638 638 21 4 31 5 21 2 3 + --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 319 335 2 4 21 3 2 536 3 3 + --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 638 319 21 4 14 4 2 78 5 + --- F(n) F(n + 1) + -- F(n) F(n + 1) + --- F(n) F(n + 1) 319 29 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 7 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 102 5 19 3 3 2 13 2 G(n) = ---- F(n) F(n + 1) + -- F(n) F(n + 1) - 3 F(n + 1) - -- F(n) 11 11 22 25 2 13 2 20 + -- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 11 127 2 4 71 6 3 13 3 + --- F(n) F(n + 1) + -- F(n + 1) - 5/22 F(n + 1) + -- F(n) 22 22 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 8 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 6 3 G(n) = -1/10 F(n + 1) + 1/6 F(n) - 1/2 - 4/3 F(n) F(n + 1) 3 3 3 4 2 + 7/6 F(n) F(n + 1) + 17/3 F(n) F(n + 1) - 7 F(n) F(n + 1) 2 4 4 + 1/10 F(n + 1) + 1/2 F(n + 1) + 1/3 F(n) + 6/5 F(n) F(n + 1) 5 - 1/5 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 9 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 G(n) = -5/22 F(n + 1) + 5/22 F(n + 1) - 2/11 F(n) F(n + 1) 2 2 2 3 + 1/22 F(n) F(n + 1) + 4/11 F(n) + 3/22 F(n) F(n + 1) - 4/11 F(n) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 10 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) G(n) = -1/12 F(n) (F(n) - 1 - 2 F(n + 1)) (F(n) + 1 - 2 F(n + 1)) (F(n) - 2 F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 11 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 105 105 2 127 84 51 2 G(n) = --- F(n + 1) - --- F(n + 1) - --- F(n) + --- F(n) F(n + 1) - --- F(n) 638 638 638 319 319 763 2 3 1227 3 2 148 4 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 638 638 319 229 5 + --- F(n) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 12 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 15 3 G(n) = 5/22 - 5/22 F(n + 1) - 4/11 F(n) F(n + 1) + -- F(n) F(n + 1) 22 2 2 2 3 + 9/11 F(n) F(n + 1) + 1/2 F(n) F(n + 1) - 9/22 F(n) 31 3 4 - -- F(n) F(n + 1) + 2/11 F(n) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 13 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = -1/12 F(n) (-11 + 5 F(n + 1) - 5 F(n) F(n + 1) + F(n) 2 2 3 4 - 35 F(n) F(n + 1) + 35 F(n) F(n + 1) + 10 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 14 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 19 3 2 2 G(n) = - 5/22 + 5/22 F(n + 1) + -- F(n) - F(n) F(n + 1) + F(n) F(n + 1) 22 53 2 3 3 27 3 2 4 + -- F(n) F(n + 1) - F(n) F(n + 1) - -- F(n) F(n + 1) + F(n) 22 22 4 18 5 - 9/22 F(n) F(n + 1) - -- F(n) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 15 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - 2 j) F(2 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 G(n) = 5/22 F(n + 1) - 5/22 F(n + 1) - 1/11 F(n) + 5/22 F(n) F(n + 1) 2 10 2 3 3 2 - 3/11 F(n) + -- F(n) F(n + 1) - 3/2 F(n) F(n + 1) 11 4 5 + 4/11 F(n) F(n + 1) + 4/11 F(n) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 16 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - 2 j) F(3 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 449 2 489 6 79 79 2 G(n) = --- F(n + 1) + --- F(n + 1) - --- F(n + 1) + --- F(n) - --- F(n) 319 638 638 638 638 38 1021 5 717 2 4 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 319 638 638 831 3 3 - --- F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 17 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - 2 j) F(3 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 15 2 3 13 3 G(n) = 3/22 F(n) F(n + 1) - -- F(n) F(n + 1) - 5/22 F(n + 1) + -- F(n) 22 22 5 41 2 4 89 3 3 + 9/11 F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 22 22 21 4 2 2 13 2 + -- F(n) F(n + 1) + 5/22 F(n + 1) - -- F(n) 11 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 18 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - 2 j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 5 3 3 G(n) = 1/84 F(n) (-95 F(n + 1) + 200 F(n) F(n + 1) - 330 F(n) F(n + 1) 4 2 2 4 + 2 F(n) + 81 F(n + 1) - 2 F(n) + 2 F(n + 1) + 4 F(n) F(n + 1) 3 2 2 2 4 - 4 F(n) F(n + 1) - 2 F(n) F(n + 1) + 140 F(n) F(n + 1) 3 + 4 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 19 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - 2 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 3 107 2 1347 4 3 1855 5 2 G(n) = ---- F(n + 1) - --- F(n) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 638 638 23 6 221 187 2 + --- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 290 638 290 819 2 20 2 107 7 28 6 - ---- F(n) F(n + 1) + --- F(n + 1) + --- F(n) - ---- F(n) F(n + 1) 3190 319 319 1595 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 20 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - 2 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 145 3 19 3 19 G(n) = 7/22 F(n) F(n + 1) + 7/11 F(n + 1) + --- F(n + 1) + -- F(n) - -- 22 22 22 37 3 41 2 23 2 2 - -- F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 22 22 11 115 2 5 39 3 6 + --- F(n) F(n + 1) + -- F(n) F(n + 1) + 25/2 F(n) F(n + 1) 11 11 485 3 4 70 7 - --- F(n) F(n + 1) - -- F(n + 1) 22 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 21 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(2 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 G(n) = -5/22 F(n + 1) + 5/22 F(n + 1) - 2/11 F(n) F(n + 1) 2 2 2 3 + 1/22 F(n) F(n + 1) + 4/11 F(n) + 3/22 F(n) F(n + 1) - 4/11 F(n) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 22 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - 2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 20 3 353 2 387 3 G(n) = --- - --- F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 638 638 465 2 25 2 2 263 3 + --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) 638 319 638 409 3 223 4 - --- F(n) F(n + 1) + --- F(n) 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 23 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - 2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = -1/84 F(n) (-46 + 34 F(n + 1) - 34 F(n) F(n + 1) + 11 F(n) 2 2 3 4 - 160 F(n) F(n + 1) + 160 F(n) F(n + 1) + 35 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 24 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - 2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 13 3 G(n) = 5/22 - 5/22 F(n + 1) - 3/11 F(n) F(n + 1) + -- F(n) F(n + 1) 22 13 2 2 2 3 + -- F(n) F(n + 1) + 2/11 F(n) F(n + 1) - 5/11 F(n) 22 19 3 4 - -- F(n) F(n + 1) + 5/22 F(n) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 25 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 20 529 575 3 G(n) = - --- + --- F(n + 1) + --- F(n) - --- F(n) F(n + 1) 319 319 638 638 719 2 2 387 2 3 845 3 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 638 279 3 2 247 4 196 4 983 5 + --- F(n) F(n + 1) + --- F(n) - --- F(n) F(n + 1) - --- F(n) 638 319 319 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 26 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 15 3 G(n) = - 5/22 + 5/22 F(n + 1) + -- F(n) - 9/11 F(n) F(n + 1) 22 2 2 2 3 19 3 + 5/11 F(n) F(n + 1) + 3/2 F(n) F(n + 1) - -- F(n) F(n + 1) 22 23 4 4 5 + -- F(n) - 1/2 F(n) F(n + 1) - 3/2 F(n) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 27 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 62 291 2 129 5 6 14 G(n) = -- F(n + 1) + --- F(n + 1) - --- F(n + 1) - 13 F(n + 1) - -- F(n) 11 22 22 11 14 2 115 133 4 5 + -- F(n) - --- F(n) F(n + 1) + --- F(n) F(n + 1) + 32 F(n) F(n + 1) 11 22 11 41 2 3 2 4 166 3 2 + -- F(n) F(n + 1) + 3/2 F(n) F(n + 1) - --- F(n) F(n + 1) 11 11 3 3 - 29 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 28 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(2 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 2 2 G(n) = -1/6 F(n) (-5 + F(n + 1) - F(n) F(n + 1) + 2 F(n) - 13 F(n) F(n + 1) 3 4 + 13 F(n) F(n + 1) + 3 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 29 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - j) F(3 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 6 2 G(n) = -5/22 F(n + 1) + 1/11 F(n) - 3/11 F(n + 1) + 1/2 F(n + 1) 2 5 83 3 3 - 1/11 F(n) - 3/11 F(n) F(n + 1) - -- F(n) F(n + 1) 22 83 2 4 2 + 7/22 F(n) F(n + 1) + -- F(n) F(n + 1) + 7/11 F(n) F(n + 1) 22 15 2 - -- F(n) F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 30 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - j) F(3 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 2 G(n) = 1/6 F(n) (-2 F(n + 1) + 6 F(n + 1) + 3 F(n) F(n + 1) 4 2 2 3 - 13 F(n) F(n + 1) - 5 F(n) F(n + 1) + 18 F(n) F(n + 1) 3 2 4 3 - 20 F(n) F(n + 1) + 13 F(n) F(n + 1) - 2 F(n) + 2 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 31 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1021 2 3 2 1855 3 4 G(n) = ----- F(n) F(n + 1) - 9/319 F(n) + 9/319 F(n) - ---- F(n) F(n + 1) 638 638 105 11 2 105 3 + --- F(n) F(n + 1) + -- F(n) F(n + 1) + --- F(n + 1) 638 58 638 875 6 1395 2 5 1585 4 3 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 638 319 105 2 - --- F(n + 1) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 32 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 145 3 4 70 7 15 2 G(n) = --- F(n + 1) - 4/11 + 3/22 F(n + 1) - -- F(n + 1) - -- F(n) F(n + 1) 22 11 22 6 485 3 4 3 + 25/2 F(n) F(n + 1) - --- F(n) F(n + 1) + 7/22 F(n) F(n + 1) 22 2 23 2 2 115 2 5 - 4/11 F(n) F(n + 1) - -- F(n) F(n + 1) + --- F(n) F(n + 1) 11 11 17 3 3 + -- F(n) F(n + 1) + 4/11 F(n) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 33 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 2 2 G(n) = -1/6 F(n) (-85 F(n) F(n + 1) + 11 F(n) F(n + 1) 3 3 6 3 + 245 F(n) F(n + 1) + 80 F(n + 1) - 6 F(n) F(n + 1) 5 4 2 - 175 F(n) F(n + 1) + 3 F(n + 1) + 22 F(n) F(n + 1) - 87 F(n + 1) 3 2 4 - 8 F(n) F(n + 1) - 4 F(n) + 4 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 34 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 13 3 G(n) = 5/22 - 5/22 F(n + 1) - 3/11 F(n) F(n + 1) + -- F(n) F(n + 1) 22 13 2 2 2 3 + -- F(n) F(n + 1) + 2/11 F(n) F(n + 1) - 5/11 F(n) 22 19 3 4 - -- F(n) F(n + 1) + 5/22 F(n) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 35 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = -1/12 F(n) (-9 + 3 F(n + 1) - 3 F(n) F(n + 1) + 2 F(n) 2 2 3 4 - 27 F(n) F(n + 1) + 27 F(n) F(n + 1) + 7 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 36 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 126 2 393 2 G(n) = -3/319 F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 638 105 3 93 3 111 2 93 2 - --- F(n + 1) + --- F(n) + --- F(n + 1) - --- F(n) 638 319 638 319 298 5 2153 3 3 397 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 319 638 638 2493 2 4 + ---- F(n) F(n + 1) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 37 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 17 10 3 G(n) = - 5/22 + 5/22 F(n + 1) + -- F(n) - -- F(n) F(n + 1) 22 11 23 2 2 18 2 3 27 3 + -- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 22 3 2 19 4 4 31 5 - 5/22 F(n) F(n + 1) + -- F(n) - 6/11 F(n) F(n + 1) - -- F(n) 22 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 38 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 187 5 239 3 3 4 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) - 1/4 F(n + 1) 12 12 2 2 6 23 + F(n) F(n + 1) + 22/3 F(n + 1) + -- F(n) F(n + 1) 12 2 4 3 2 - 23/4 F(n) F(n + 1) - 5/4 F(n) F(n + 1) + 1/4 - 22/3 F(n + 1) 2 - 1/4 F(n) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 39 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 12 2 12 5 G(n) = -5/22 F(n + 1) + 5/22 F(n + 1) + -- F(n) - -- F(n) 11 11 12 4 21 5 37 2 3 - -- F(n) F(n + 1) + -- F(n) F(n + 1) + -- F(n) F(n + 1) 11 22 22 23 2 4 75 3 2 14 3 3 - -- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 22 11 73 4 65 4 2 51 5 + -- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 22 22 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 40 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(3 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 55 3 3 41 2 G(n) = --- F(n + 1) - 4/21 F(n) + 4/21 F(n) + -- F(n) F(n + 1) 14 28 19 2 65 2 5 755 6 - -- F(n) F(n + 1) - -- F(n) F(n + 1) - --- F(n) F(n + 1) 42 42 84 200 3 4 55 7 + --- F(n) F(n + 1) + -- F(n + 1) 21 14 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 41 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - 3 j) F(3 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 355 6 427 2 17 2 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 1775 2 5 135 3 4 49 5 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 319 1585 5 2 20 3 105 3 105 2 - ---- F(n) F(n + 1) - --- F(n + 1) - --- F(n) + --- F(n) 638 319 638 638 98 2 4 49 3 3 98 4 2 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 319 49 5 20 2 - --- F(n) F(n + 1) + --- F(n + 1) 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 42 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - 3 j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2517 15205 4 4 7907 6 2 3897 4 G(n) = - ----- + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n + 1) 12122 12122 6061 12122 16450 5 3 1530 7 9165 3 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 6061 12122 5347 2 2 61 7 2517 690 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) - ---- F(n + 1) 12122 6061 12122 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 43 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - 3 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 2 G(n) = -1/84 F(n) (-14 F(n + 1) + 625 F(n + 1) + 220 F(n) F(n + 1) 2 3 3 - 26 F(n) F(n + 1) + 7 F(n) - 7 F(n) - 655 F(n + 1) 6 3 4 2 5 - 1425 F(n) F(n + 1) + 1625 F(n) F(n + 1) - 350 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 44 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - 3 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 3 20 4725 3 5 1211 2 2 G(n) = --- F(n + 1) - --- - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 638 638 3 5809 3 427 2 - 1/2 F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 638 638 25 2 16425 4 4 275 7 - -- F(n) F(n + 1) + ----- F(n) F(n + 1) + --- F(n) F(n + 1) 58 638 29 13025 2 6 4 13 3 - ----- F(n) F(n + 1) + 3/29 F(n) - --- F(n) 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 45 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(3 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) G(n) = -1/12 F(n) (F(n) - 2 F(n + 1)) (F(n) - 1 - 2 F(n + 1)) (F(n) + 1 - 2 F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 46 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - 3 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 20 3 353 2 387 3 G(n) = --- - --- F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 638 638 465 2 25 2 2 263 3 + --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) 638 319 638 409 3 223 4 - --- F(n) F(n + 1) + --- F(n) 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 47 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - 3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 690 690 9631 10899 3 G(n) = - ---- + ---- F(n + 1) + ----- F(n) - ----- F(n) F(n + 1) 6061 6061 12122 12122 6276 2 2 9742 2 3 15009 3 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 6061 12122 1941 3 2 9897 4 293 4 9074 5 - ----- F(n) F(n + 1) + ----- F(n) - --- F(n) F(n + 1) - ---- F(n) 12122 12122 638 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 48 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - 3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = -1/12 F(n) (-9 + 3 F(n + 1) - 3 F(n) F(n + 1) + 2 F(n) 2 2 3 4 - 27 F(n) F(n + 1) + 27 F(n) F(n + 1) + 7 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 49 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - 3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 20 262 285 3 G(n) = - --- + --- F(n + 1) + --- F(n) - --- F(n) F(n + 1) 319 319 319 319 73 2 2 925 2 3 943 3 + -- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 58 638 638 23 3 2 487 4 301 4 971 5 + --- F(n) F(n + 1) + --- F(n) - --- F(n) F(n + 1) - --- F(n) 319 638 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 50 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 6 3 3 G(n) = -1/4 F(n + 1) + 23/3 F(n + 1) + 41/2 F(n) F(n + 1) 5 3 - 33/2 F(n) F(n + 1) + 2 F(n) F(n + 1) + 1/6 F(n) F(n + 1) 2 2 2 4 3 + 3/4 F(n) F(n + 1) - 11/2 F(n) F(n + 1) - 7/6 F(n) F(n + 1) + 1/4 2 2 - 23/3 F(n + 1) - 1/4 F(n) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 51 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 639 2 3 1003 2 4 1217 3 2 G(n) = --- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 319 1253 3 3 2159 4 980 4 2 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 638 638 319 3167 5 707 4 661 5 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 638 638 638 20 20 2 355 2 355 5 - --- F(n + 1) + --- F(n + 1) + --- F(n) - --- F(n) 319 319 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 52 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 25 5 6 3 4 G(n) = --- F(n + 1) - 95 F(n) F(n + 1) + 575/6 F(n) F(n + 1) - 5/12 F(n) 12 25 2 4 + -- F(n + 1) + 57/4 F(n) F(n + 1) + 25/6 F(n) F(n + 1) 12 41 2 25 2 3 2 5 - -- F(n) F(n + 1) + -- F(n) F(n + 1) - 35/3 F(n) F(n + 1) 12 12 3 2 3 3 7 - 25/4 F(n) F(n + 1) - 245/6 F(n + 1) + 5/12 F(n) + 245/6 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 53 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - 2 j) F(3 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 70 7 4 6 G(n) = - 4/11 - -- F(n + 1) + 3/22 F(n + 1) + 25/2 F(n) F(n + 1) 11 485 3 4 15 2 35 2 2 - --- F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 22 115 2 5 45 3 3 + --- F(n) F(n + 1) + -- F(n) F(n + 1) - 2/11 F(n) F(n + 1) 11 22 2 145 3 3 - 2/11 F(n) F(n + 1) + --- F(n + 1) + 4/11 F(n) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 54 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - 2 j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 5 3 G(n) = -1/84 F(n) (7 F(n) + 14 F(n + 1) - 350 F(n) F(n + 1) - 627 F(n + 1) 2 2 7 + 178 F(n) F(n + 1) - 40 F(n) F(n + 1) + 625 F(n + 1) 6 3 4 - 1425 F(n) F(n + 1) + 1625 F(n) F(n + 1) - 7 F(n)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 55 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - 2 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 13 1700 2 6 20 3 13 3 G(n) = --- + ---- F(n) F(n + 1) + --- F(n + 1) - --- F(n) 319 319 319 319 3 2 16425 8 - 6/29 F(n) F(n + 1) + 2/29 F(n) F(n + 1) - ----- F(n + 1) 638 19450 7 37575 3 5 16359 4 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n + 1) 319 638 638 2998 3 211 2 798 2 2 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 56 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - 2 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 925 4 4 575 3 5 4 G(n) = --- F(n) F(n + 1) - --- F(n) F(n + 1) + 5/22 F(n + 1) 22 22 73 4 4 18 2 2 - -- F(n) F(n + 1) - 5/11 F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 27 2 3 51 3 - -- F(n) F(n + 1) + -- F(n) F(n + 1) + 5/11 F(n) - 5/22 F(n + 1) 22 22 175 2 6 51 3 2 4 - --- F(n) F(n + 1) + -- F(n) F(n + 1) - 5/11 F(n) 11 11 116 3 225 7 - --- F(n) F(n + 1) + --- F(n) F(n + 1) 11 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 57 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(3 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 15 3 G(n) = 5/22 - 5/22 F(n + 1) - 4/11 F(n) F(n + 1) + -- F(n) F(n + 1) 22 2 2 2 3 + 9/11 F(n) F(n + 1) + 1/2 F(n) F(n + 1) - 9/22 F(n) 31 3 4 - -- F(n) F(n + 1) + 2/11 F(n) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 58 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - 2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 20 262 285 3 G(n) = - --- + --- F(n + 1) + --- F(n) - --- F(n) F(n + 1) 319 319 319 319 73 2 2 925 2 3 943 3 + -- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 58 638 638 23 3 2 487 4 301 4 971 5 + --- F(n) F(n + 1) + --- F(n) - --- F(n) F(n + 1) - --- F(n) 319 638 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 59 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - 2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 164 2 2 473 5 G(n) = -5/21 F(n + 1) - --- F(n + 1) - 5/21 F(n) - --- F(n) F(n + 1) 21 28 3 3 164 6 169 + 83/4 F(n) F(n + 1) + --- F(n + 1) + --- F(n) F(n + 1) 21 84 3 95 2 2 151 2 4 - 5/42 F(n) F(n + 1) + -- F(n) F(n + 1) - --- F(n) F(n + 1) 84 28 3 - 5/4 F(n) F(n + 1) + 5/21 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 60 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - 2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 17 10 3 G(n) = - 5/22 + 5/22 F(n + 1) + -- F(n) - -- F(n) F(n + 1) 22 11 23 2 2 18 2 3 27 3 + -- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 22 3 2 19 4 4 31 5 - 5/22 F(n) F(n + 1) + -- F(n) - 6/11 F(n) F(n + 1) - -- F(n) 22 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 61 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 347 5 347 2 20 20 2 G(n) = ---- F(n) + --- F(n) - --- F(n + 1) + --- F(n + 1) 319 319 319 319 356 4 1618 5 333 5 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 1137 2 3 1091 2 4 2209 3 2 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 638 638 700 3 3 2071 4 987 4 2 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 319 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 62 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 113 251 4 5 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) + 32 F(n) F(n + 1) 22 22 109 2 3 2 4 173 3 2 + --- F(n) F(n + 1) + 3/2 F(n) F(n + 1) - --- F(n) F(n + 1) 22 11 3 3 60 291 2 125 5 - 29 F(n) F(n + 1) + -- F(n + 1) + --- F(n + 1) - --- F(n + 1) 11 22 22 6 23 23 2 - 13 F(n + 1) - -- F(n) + -- F(n) 22 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 63 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 37 7 25 3 1561 5 37 2 G(n) = 5/2 F(n + 1) - -- F(n) - -- F(n + 1) + ---- F(n) F(n + 1) + -- F(n) 22 11 22 22 201 6 217 679 2 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n + 1) 22 22 22 46 6 247 2 4 19 2 5 - -- F(n) F(n + 1) + --- F(n) F(n + 1) + -- F(n) F(n + 1) 11 22 22 342 6 75 2 1647 3 3 - --- F(n + 1) - -- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 22 22 2 + 3/22 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 64 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(3 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 6 G(n) = 1/6 F(n) (-1 - 12 F(n + 1) + 19 F(n + 1) + 27 F(n) F(n + 1) 3 5 2 2 + 4 F(n) F(n + 1) - 72 F(n) F(n + 1) - 15 F(n) F(n + 1) 2 4 3 5 4 + 85 F(n) F(n + 1) + 13 F(n) F(n + 1) - 47 F(n) F(n + 1) - 2 F(n + 1) 6 + F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 65 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 16425 8 3143 3 153 2 G(n) = 3/638 F(n) - ----- F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 638 769 2 2 37575 3 5 189 2 + --- F(n) F(n + 1) - 3/638 - ----- F(n) F(n + 1) + --- F(n) F(n + 1) 319 638 638 19450 7 105 3 3 + ----- F(n) F(n + 1) - --- F(n + 1) - 8/319 F(n) F(n + 1) 319 638 1503 4 1700 2 6 + ---- F(n + 1) + ---- F(n) F(n + 1) 58 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 66 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 233 3 225 7 175 2 6 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 22 22 11 925 4 4 13 4 575 3 5 21 + --- F(n) F(n + 1) - -- F(n) + 5/22 - --- F(n) F(n + 1) + -- F(n) 22 11 22 22 4 2 2 - 5/22 F(n + 1) - 5/11 F(n) F(n + 1) - 9/22 F(n) F(n + 1) 2 3 41 3 91 3 2 - 8/11 F(n) F(n + 1) + -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 22 42 4 - -- F(n) F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 67 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 2 4 G(n) = 1175/6 F(n + 1) - 1/2 + 1/2 F(n) - 586/3 F(n + 1) 3 3 2 6 2 2 - 101/3 F(n) F(n + 1) - 25/2 F(n) F(n + 1) - 55/2 F(n) F(n + 1) 3 5 2 4 - 11/3 F(n) F(n + 1) + 425 F(n) F(n + 1) + 7 F(n) F(n + 1) 7 3 2 - 2825/6 F(n) F(n + 1) + 159/2 F(n) F(n + 1) + 79/6 F(n + 1) 3 5 6 + 41/6 F(n) F(n + 1) + 88/3 F(n) F(n + 1) - 79/6 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 68 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 15 3 G(n) = - 5/22 + 5/22 F(n + 1) + -- F(n) - 9/11 F(n) F(n + 1) 22 2 2 2 3 19 3 + 5/11 F(n) F(n + 1) + 3/2 F(n) F(n + 1) - -- F(n) F(n + 1) 22 23 4 4 5 + -- F(n) - 1/2 F(n) F(n + 1) - 3/2 F(n) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 69 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 4 2 2 G(n) = -7/6 F(n) F(n + 1) - 1/4 F(n + 1) - 1/4 F(n) - 23/3 F(n + 1) + 1/4 5 3 3 3 - 33/2 F(n) F(n + 1) + 41/2 F(n) F(n + 1) + 1/6 F(n) F(n + 1) 2 2 2 4 6 + 3/4 F(n) F(n + 1) - 11/2 F(n) F(n + 1) + 23/3 F(n + 1) + 2 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 70 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 205 3 4 63 3 362 2 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 22 319 319 489 2 2 1041 3 6 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) - 5/22 F(n) F(n + 1) 319 638 45 2 5 70 4 3 105 3 144 3 + -- F(n) F(n + 1) + -- F(n) F(n + 1) + --- F(n + 1) + --- F(n) 11 11 638 319 183 4 75 2 105 - --- F(n) - --- F(n) F(n + 1) - --- 638 319 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 71 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 23 2 23 5 4 G(n) = -5/22 F(n + 1) + 5/22 F(n + 1) + -- F(n) - -- F(n) - F(n) F(n + 1) 22 22 19 5 15 2 3 27 2 4 + -- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 22 73 3 2 47 3 3 79 4 - -- F(n) F(n + 1) + -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 22 22 30 4 2 113 5 + -- F(n) F(n + 1) - --- F(n) F(n + 1) 11 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 72 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 5 G(n) = -1/12 F(n) (43 F(n) F(n + 1) + 160 F(n + 1) - 350 F(n) F(n + 1) 3 3 3 2 2 + 490 F(n) F(n + 1) - 14 F(n) F(n + 1) + 26 F(n) F(n + 1) 2 4 3 4 4 - 170 F(n) F(n + 1) - 19 F(n) F(n + 1) + 7 F(n + 1) + 4 F(n) 2 2 - 4 F(n) - 173 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 73 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 4 2 291 2 41 2 G(n) = 5/22 F(n + 1) + 61/2 F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) 11 22 41 3 213 6 201 2 - -- F(n) - --- F(n) F(n + 1) + 55/2 F(n) F(n + 1) + --- F(n) F(n + 1) 22 22 22 2 5 3 3 3 4 - 125/2 F(n) F(n + 1) - 13 F(n) F(n + 1) - 5 F(n) F(n + 1) 4 3 2 4 5 + 60 F(n) F(n + 1) - 19 F(n) F(n + 1) + 17/2 F(n) F(n + 1) 2 - 5/22 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 74 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 4 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 678 678 2 262281 4 128245 8 G(n) = ---- F(n) - ---- F(n) - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 6061 6061 12122 6061 303965 2 7 64565 3 6 20795 4 5 - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 12122 418 183 4235 2 3 381 3 2 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 12122 551 209 1988 4 690 2 690 9 + ---- F(n) F(n + 1) - ---- F(n + 1) + ---- F(n + 1) 6061 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 75 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 4 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 31 3139 4 2 G(n) = 5/84 F(n) - -- F(n + 1) + ---- F(n) F(n + 1) + 5/14 F(n) F(n + 1) 42 84 2 3 3725 9 2 3 - 46/3 F(n) F(n + 1) + ---- F(n + 1) - 5/14 F(n) F(n + 1) - 5/84 F(n) 42 1847 5 3025 8 125 2 7 - ---- F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 21 14 42 79 3 2 2225 3 6 + -- F(n) F(n + 1) + ---- F(n) F(n + 1) 14 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 76 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 4 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 60 4 15 15 149 303 3 G(n) = ---- F(n + 1) + --- - --- F(n) + --- F(n + 1) + --- F(n) F(n + 1) 319 638 638 638 638 187125 8 82325 3 6 249 2 2 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 319 5807 2 3 1390 3 2 16338 4 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 319 77181 5 77225 9 1325 2 7 + ----- F(n + 1) - ----- F(n + 1) + ---- F(n) F(n + 1) 638 638 638 126 3 + --- F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 77 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 105 105 2 127 84 51 2 G(n) = --- F(n + 1) - --- F(n + 1) - --- F(n) + --- F(n) F(n + 1) - --- F(n) 638 638 638 319 319 763 2 3 1227 3 2 148 4 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 638 638 319 229 5 + --- F(n) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 78 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 4 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = -1/84 F(n) (-46 + 34 F(n + 1) - 34 F(n) F(n + 1) + 11 F(n) 2 2 3 4 - 160 F(n) F(n + 1) + 160 F(n) F(n + 1) + 35 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 79 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 4 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 690 690 9631 10899 3 G(n) = - ---- + ---- F(n + 1) + ----- F(n) - ----- F(n) F(n + 1) 6061 6061 12122 12122 6276 2 2 9742 2 3 15009 3 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 6061 12122 1941 3 2 9897 4 293 4 9074 5 - ----- F(n) F(n + 1) + ----- F(n) - --- F(n) F(n + 1) - ---- F(n) 12122 12122 638 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 80 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 4 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 1693 2 4 1269 3 3 400 4 2 G(n) = ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 638 319 319 211 5 397 5 93 2 105 2 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) + --- F(n + 1) 638 638 319 638 105 3 93 3 126 2 393 2 - --- F(n + 1) + --- F(n) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 319 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 81 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 3 G(n) = 1/84 F(n) (-20 F(n) + 20 F(n) + 262 F(n) F(n + 1) 3 2 4 3 5 - 318 F(n) F(n + 1) + 169 F(n) F(n + 1) - 50 F(n + 1) + 62 F(n + 1) 2 4 2 + 75 F(n) F(n + 1) - 135 F(n) F(n + 1) - 65 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 82 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 4 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 32499 5441 2 33189 5 14219 6 G(n) = ----- F(n + 1) + ---- F(n + 1) - ----- F(n + 1) - ----- F(n + 1) 6061 418 6061 1102 6672 6672 2 6993 4 17498 5 - ---- F(n) + ---- F(n) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 6061 638 551 29397 2 3 1707 2 4 92727 3 2 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 1102 6061 31797 3 3 3167 - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 1102 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 83 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 183 4 75 2 144 3 205 3 4 G(n) = ---- F(n) - --- F(n) F(n + 1) + --- F(n) - --- F(n) F(n + 1) 638 319 319 22 1041 3 63 3 362 2 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 319 489 2 2 105 3 105 70 4 3 - --- F(n) F(n + 1) + --- F(n + 1) - --- + -- F(n) F(n + 1) 319 638 638 11 45 2 5 6 + -- F(n) F(n + 1) - 5/22 F(n) F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 84 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 4 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 G(n) = -1/84 F(n) (-1198 F(n + 1) + 288 F(n) F(n + 1) + 31 F(n) 6 2 4 3 3 + 1120 F(n + 1) - 1190 F(n) F(n + 1) + 3430 F(n) F(n + 1) 4 3 2 2 + 66 F(n + 1) - 132 F(n) F(n + 1) + 194 F(n) F(n + 1) 3 5 2 - 128 F(n) F(n + 1) - 2450 F(n) F(n + 1) - 31 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 85 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 4 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 1726 105 1314 3 2 702 4 G(n) = ---- - --- F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 319 319 11739 4 1963 4 1424 2 2 + ----- F(n + 1) - ---- F(n) + ---- F(n) F(n + 1) 319 319 319 876 2 3 5393 3 575 2 6 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 319 638 22 925 8 2075 7 2425 3 5 237 5 - --- F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) 22 22 22 319 105 5 + --- F(n + 1) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 86 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 31 2 1039 5 4 6 G(n) = -- F(n) F(n + 1) + ---- F(n) F(n + 1) + 5/84 F(n) F(n + 1) 84 84 19 2 5 1739 4 5 25 6 + -- F(n) F(n + 1) - ---- F(n) F(n + 1) - -- F(n) F(n + 1) 42 252 42 73 6 3 7 2 499 8 - -- F(n) F(n + 1) - 1/84 F(n) F(n + 1) - --- F(n) F(n + 1) 21 252 3 47 5 47 9 19 7 + 1/7 F(n + 1) - -- F(n + 1) + --- F(n + 1) - -- F(n) 90 252 84 5 2 7 214 47 - 2/7 F(n) F(n + 1) - 1/7 F(n + 1) + --- F(n) + --- F(n + 1) 315 140 571 5 - ---- F(n) 1260 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 87 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 3 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 77225 4 5 16338 4 179 2 2 G(n) = ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 638 5807 2 3 1390 3 2 94 271 3 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- + --- F(n) F(n + 1) 319 319 319 319 94 5 5100 3 6 + --- F(n) + 2/319 F(n + 1) - 2/29 F(n + 1) - ---- F(n) F(n + 1) 319 319 114 4 509 3 32675 8 + --- F(n + 1) - --- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 638 638 3450 2 7 - ---- F(n) F(n + 1) 29 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 88 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = -1/12 F(n) (-11 + 5 F(n + 1) - 5 F(n) F(n + 1) + F(n) 2 2 3 4 - 35 F(n) F(n + 1) + 35 F(n) F(n + 1) + 10 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 89 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 3 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 20 529 575 3 G(n) = - --- + --- F(n + 1) + --- F(n) - --- F(n) F(n + 1) 319 319 638 638 719 2 2 387 2 3 845 3 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 638 279 3 2 247 4 196 4 983 5 + --- F(n) F(n + 1) + --- F(n) - --- F(n) F(n + 1) - --- F(n) 638 319 319 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 90 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6672 6672 2 32499 5441 2 G(n) = ----- F(n) + ---- F(n) + ----- F(n + 1) + ---- F(n + 1) 6061 6061 6061 418 33189 5 14219 6 3167 - ----- F(n + 1) - ----- F(n + 1) - ---- F(n) F(n + 1) 6061 1102 638 6993 4 17498 5 29397 2 3 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 638 551 6061 1707 2 4 92727 3 2 31797 3 3 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 1102 6061 1102 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 91 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 23 2 4 G(n) = -22/3 F(n + 1) - 1/4 F(n) + -- F(n) F(n + 1) - 23/4 F(n) F(n + 1) 12 3 4 2 2 - 5/4 F(n) F(n + 1) - 1/4 F(n + 1) + F(n) F(n + 1) 187 5 239 3 3 6 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + 1/4 + 22/3 F(n + 1) 12 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 92 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3419 4243 2 3459 5 4223 6 G(n) = ---- F(n + 1) + ---- F(n + 1) - ---- F(n + 1) - ---- F(n + 1) 638 319 638 319 347 347 2 1618 3450 4 - --- F(n) + --- F(n) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 319 319 10397 5 1604 2 3 883 2 4 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 638 9821 3 2 9364 3 3 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 93 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 173 2 3 3 G(n) = -9/4 F(n + 1) + --- F(n) F(n + 1) + 1/3 F(n) - 245/6 F(n + 1) 12 43 2 2 3 - 1/3 F(n) + 9/4 F(n + 1) - -- F(n) F(n + 1) + 11/4 F(n) F(n + 1) 12 2 5 3 2 4 - 35/3 F(n) F(n + 1) - 7 F(n) F(n + 1) + 17/4 F(n) F(n + 1) 3 4 7 6 + 575/6 F(n) F(n + 1) + 245/6 F(n + 1) - 95 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 94 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 2 574 2 2892 2 2 5 G(n) = ---- F(n + 1) + --- F(n) + ---- F(n) F(n + 1) - 125/2 F(n) F(n + 1) 319 319 319 4188 3 3 20 3 574 3 3 4 - ---- F(n) F(n + 1) + --- F(n + 1) - --- F(n) - 5 F(n) F(n + 1) 319 319 319 4 3 3093 6 + 60 F(n) F(n + 1) - ---- F(n) F(n + 1) + 55/2 F(n) F(n + 1) 319 761 2 5337 5 5934 2 4 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 29 638 319 19291 4 2 + ----- F(n) F(n + 1) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 95 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 5 2 6 G(n) = 1175/6 F(n + 1) + 121/4 F(n) F(n + 1) - 25/2 F(n) F(n + 1) - 7/12 3 6 47 + 475/6 F(n) F(n + 1) - 27/2 F(n + 1) - -- F(n) F(n + 1) 12 7 2 2 2 4 - 2825/6 F(n) F(n + 1) - 82/3 F(n) F(n + 1) + 27/4 F(n) F(n + 1) 3 5 4 2 2 + 425 F(n) F(n + 1) - 781/4 F(n + 1) + 27/2 F(n + 1) + 7/12 F(n) 85 3 3 3 + -- F(n) F(n + 1) - 137/4 F(n) F(n + 1) 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 96 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 2 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 29 2 29 151 2503 5 578 5 G(n) = --- F(n) + -- F(n) - --- F(n + 1) + ---- F(n + 1) - --- F(n) F(n + 1) 22 22 22 11 11 575 2 7 208 3 2 2425 9 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n + 1) 11 11 11 509 2 1036 4 1257 3 3 - --- F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 22 11 22 5600 3 6 157 11425 8 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) + ----- F(n) F(n + 1) 11 22 22 130 2 3 237 2 4 252 6 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n + 1) 11 22 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 97 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 19 3 2 2 G(n) = - 5/22 + 5/22 F(n + 1) + -- F(n) - F(n) F(n + 1) + F(n) F(n + 1) 22 53 2 3 3 27 3 2 4 + -- F(n) F(n + 1) - F(n) F(n + 1) - -- F(n) F(n + 1) + F(n) 22 22 4 18 5 - 9/22 F(n) F(n + 1) - -- F(n) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 98 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3167 651 4 5 G(n) = ----- F(n) F(n + 1) + --- F(n) F(n + 1) + 32 F(n) F(n + 1) 638 58 3437 2 3 2 4 5151 3 2 + ---- F(n) F(n + 1) + 3/2 F(n) F(n + 1) - ---- F(n) F(n + 1) 638 319 3 3 3539 4167 2 3579 5 - 29 F(n) F(n + 1) + ---- F(n + 1) + ---- F(n + 1) - ---- F(n + 1) 638 319 638 6 355 355 2 - 13 F(n + 1) - --- F(n) + --- F(n) 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 99 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 599 2 7 31 3 3 G(n) = --- F(n) F(n + 1) + 245/6 F(n + 1) + -- F(n) - 245/6 F(n + 1) 42 84 95 5 31 95 6 - -- F(n + 1) - -- F(n) + -- F(n + 1) - 95 F(n) F(n + 1) 42 84 42 3 4 115 4 2 + 575/6 F(n) F(n + 1) + --- F(n) F(n + 1) - 24/7 F(n) F(n + 1) 28 65 2 3 2 5 605 3 2 + -- F(n) F(n + 1) - 35/3 F(n) F(n + 1) - --- F(n) F(n + 1) 21 84 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 100 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 21 2 4 3 2 G(n) = -1/2 F(n) F(n + 1) + -- F(n) F(n + 1) - 9/2 F(n) F(n + 1) 11 306 3 3 4 51 - --- F(n) F(n + 1) + 11/2 F(n) F(n + 1) - -- F(n) F(n + 1) 11 11 661 5 12 2 137 2 + --- F(n) F(n + 1) + -- F(n) - 5/22 F(n + 1) + --- F(n + 1) 22 11 11 269 6 12 - --- F(n + 1) - -- F(n) 22 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 101 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 574 6 5312 6 492 2 2479 3 574 7 G(n) = --- F(n) + ---- F(n + 1) - --- F(n + 1) + ---- F(n + 1) - --- F(n) 319 1595 145 3190 319 2279 7 10316 5 149 6 - ---- F(n + 1) - ----- F(n) F(n + 1) + --- F(n) F(n + 1) 3190 1595 58 189 2 7423 4 2 701 4 3 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 1595 638 638 10254 5 2 1744 6 485 5 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 1595 319 58 591 + ---- F(n) F(n + 1) 1595 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 102 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 17 3 4 4 2 27 4 3 G(n) = -- F(n) F(n + 1) + 23/2 F(n) F(n + 1) + -- F(n) F(n + 1) 11 22 72 5 2 119 6 41 6 3 - -- F(n) F(n + 1) + --- F(n) F(n + 1) + -- F(n) + 5/22 F(n + 1) 11 22 22 2 49 5 + 5/22 F(n) F(n + 1) - -- F(n) F(n + 1) - 3/44 F(n) F(n + 1) 44 6 359 3 3 213 5 + 9/11 F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 44 44 2 41 7 - 5/22 F(n + 1) - -- F(n) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 103 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 7 227 5 2 65 7 G(n) = 5/22 F(n + 1) - 21/2 F(n) F(n + 1) - --- F(n) F(n + 1) - -- F(n) 11 22 345 6 2 14 6 3 5 + --- F(n) F(n + 1) - -- F(n) F(n + 1) + 5 F(n) F(n + 1) 22 11 15 4 4 115 5 3 257 6 65 8 - -- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) + -- F(n) 22 11 22 22 457 4 3 23 2 6 261 3 4 + --- F(n) F(n + 1) - -- F(n) F(n + 1) - --- F(n) F(n + 1) 22 22 22 29 2 5 3 15 7 + -- F(n) F(n + 1) - 5/22 F(n + 1) + -- F(n) F(n + 1) 11 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 104 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 3 9 G(n) = -25 F(n) F(n + 1) - 7/6 F(n) + 135/2 F(n + 1) + 2650/3 F(n + 1) 6 2 7 3 2 + 955/6 F(n) F(n + 1) + 125/2 F(n) F(n + 1) + 223/3 F(n) F(n + 1) 3 4 7 4 - 925/6 F(n) F(n + 1) - 135/2 F(n + 1) + 2231/6 F(n) F(n + 1) 8 2 2 3 - 4325/2 F(n) F(n + 1) + 20/3 F(n) F(n + 1) - 167 F(n) F(n + 1) 2 5 3 6 5 + 40/3 F(n) F(n + 1) + 10925/6 F(n) F(n + 1) - 5227/6 F(n + 1) + 7/6 F(n) - 73/6 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 105 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 129 5 6 14 14 2 62 G(n) = ---- F(n + 1) - 13 F(n + 1) - -- F(n) + -- F(n) + -- F(n + 1) 22 11 11 11 291 2 115 133 4 + --- F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 22 22 11 5 41 2 3 2 4 + 32 F(n) F(n + 1) + -- F(n) F(n + 1) + 3/2 F(n) F(n + 1) 11 166 3 2 3 3 - --- F(n) F(n + 1) - 29 F(n) F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 106 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 5 G(n) = 5/12 F(n) + 5/12 F(n + 1) - 5/6 F(n) - 5/12 F(n + 1) 41 2 11 7 3 11 3 - -- F(n) F(n + 1) + -- F(n + 1) + 5/12 F(n) - -- F(n + 1) 12 24 24 25 2 3 2 5 25 3 2 + -- F(n) F(n + 1) + 5/24 F(n) F(n + 1) - -- F(n) F(n + 1) 12 12 4 3 5 2 3 4 - 95/8 F(n) F(n + 1) + 57/4 F(n) F(n + 1) + 5/6 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 107 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 237 5 2425 3 5 3926 3 G(n) = --- F(n) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 22 319 2075 7 13702 4 925 8 + ---- F(n) F(n + 1) + ----- F(n + 1) - --- F(n + 1) 22 319 22 575 2 6 237 2459 3 1509 5 + --- F(n) F(n + 1) - --- + ---- F(n) F(n + 1) + ---- F(n + 1) 22 319 638 638 2718 3 2 1404 4 49 2 2 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - -- F(n) F(n + 1) 319 319 29 1578 2 3 807 - ---- F(n) F(n + 1) - --- F(n + 1) 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 108 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 37 3 100 2 2 5 G(n) = 5/22 F(n + 1) - -- F(n) + --- F(n) F(n + 1) - 125/2 F(n) F(n + 1) 22 11 279 3 3 3 4 4 3 - --- F(n) F(n + 1) - 5 F(n) F(n + 1) + 60 F(n) F(n + 1) 22 193 5 2 37 2 292 2 + --- F(n) F(n + 1) - 5/22 F(n + 1) + -- F(n) - --- F(n) F(n + 1) 22 22 11 6 217 437 2 4 + 55/2 F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 22 22 342 4 2 + --- F(n) F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 109 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 2 5 3 4 G(n) = -1/12 F(n) (-39 F(n + 1) - 400 F(n) F(n + 1) + 2350 F(n) F(n + 1) 7 2 6 + 47 F(n + 1) + 950 F(n + 1) + 321 F(n) F(n + 1) - 2200 F(n) F(n + 1) 2 2 3 3 - 71 F(n) F(n + 1) + 87 F(n) F(n + 1) - 964 F(n + 1) 4 3 2 3 + 81 F(n) F(n + 1) - 162 F(n) F(n + 1) - 7 F(n) + 7 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 110 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 3 4 8 G(n) = 715/2 F(n) F(n + 1) - 665/2 F(n) F(n + 1) - 525/2 F(n + 1) 2 6 5715 4 491 3 - 75/2 F(n) F(n + 1) + ---- F(n + 1) - --- F(n) F(n + 1) 22 22 1293 2 7 65 65 3 - ---- F(n) F(n + 1) - 150 F(n + 1) + -- - -- F(n) 22 22 22 2595 3 7 387 2 - ---- F(n) F(n + 1) + 650 F(n) F(n + 1) + --- F(n) F(n + 1) 22 22 1197 2 2 2 5 3 5 + ---- F(n) F(n + 1) + 35/2 F(n) F(n + 1) - 525 F(n) F(n + 1) 22 3295 3 + ---- F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 111 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = -1/6 F(n) (-1 - 3 F(n + 1) + 3 F(n) F(n + 1) + F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 112 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 G(n) = 5/22 - 5/22 F(n + 1) + 7/22 F(n) - 2/11 F(n) F(n + 1) 23 2 2 3 4 + -- F(n) F(n + 1) - 7/11 F(n) F(n + 1) - 6/11 F(n) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 113 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 105 105 2 21 2 21 5 G(n) = ---- F(n + 1) + --- F(n + 1) - --- F(n) + --- F(n) 638 638 638 638 21 3 2 536 3 3 21 4 - --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 638 319 319 14 4 2 78 5 21 4 + -- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 29 319 638 31 5 21 2 3 335 2 4 - --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 114 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) G(n) = 1/3 F(n + 1) F(n) (F(n) - F(n + 1)) (F(n) - 2 F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 115 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 13 G(n) = 5/22 F(n + 1) - 5/22 F(n + 1) - -- F(n) + 8/11 F(n) F(n + 1) 22 2 31 2 3 3 2 - 3/11 F(n) + -- F(n) F(n + 1) - 5/2 F(n) F(n + 1) 22 4 19 5 + 4/11 F(n) F(n + 1) + -- F(n) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 116 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 2 2 G(n) = -1/6 F(n) (-5 + F(n + 1) - F(n) F(n + 1) - F(n) - 16 F(n) F(n + 1) 3 4 + 16 F(n) F(n + 1) + 6 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 117 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 102 5 19 3 3 3 13 3 G(n) = ---- F(n) F(n + 1) + -- F(n) F(n + 1) - 5/22 F(n + 1) + -- F(n) 11 11 22 20 127 2 4 13 2 2 + -- F(n) F(n + 1) + --- F(n) F(n + 1) - -- F(n) - 3 F(n + 1) 11 22 22 71 6 25 2 13 2 + -- F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 22 22 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 118 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 2 G(n) = 1/6 F(n) (-2 F(n + 1) + 6 F(n + 1) + 3 F(n) F(n + 1) 4 2 2 3 - 16 F(n) F(n + 1) + F(n) F(n + 1) + 30 F(n) F(n + 1) 3 2 4 3 - 26 F(n) F(n + 1) + 4 F(n) F(n + 1) + F(n) - F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 119 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 79 2 20 449 2 489 6 79 G(n) = ---- F(n) + --- F(n + 1) + --- F(n + 1) - --- F(n + 1) + --- F(n) 638 319 638 638 638 38 1021 5 717 2 4 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 319 638 638 831 3 3 - --- F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 120 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 5 G(n) = -1/84 F(n) (-2 F(n) + 4 F(n) F(n + 1) + 126 F(n) F(n + 1) 2 2 3 4 2 + 2 F(n) F(n + 1) - 4 F(n) F(n + 1) - 140 F(n) F(n + 1) 5 4 2 + 14 F(n) F(n + 1) - 2 F(n + 1) - 14 F(n) F(n + 1) + 77 F(n + 1) 6 6 + 2 F(n) - 63 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 121 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 G(n) = 5/22 F(n + 1) - 5/22 F(n + 1) - 1/11 F(n) + 5/22 F(n) F(n + 1) 2 10 2 3 3 2 - 3/11 F(n) + -- F(n) F(n + 1) - 3/2 F(n) F(n + 1) 11 4 5 + 4/11 F(n) F(n + 1) + 4/11 F(n) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 122 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 221 17 2 1775 2 5 G(n) = --- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 319 638 107 3 1565 3 1585 7 7215 6 + --- F(n) + ---- F(n + 1) - ---- F(n + 1) + ---- F(n) F(n + 1) 319 319 319 638 745 3 4 525 2 20 2 107 2 - --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n + 1) - --- F(n) 58 319 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 123 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 32 5 13 3 14 2 2 G(n) = --- F(n) F(n + 1) + -- F(n) + -- F(n) + 3/22 F(n) F(n + 1) 11 22 11 15 2 3 3 83 4 2 41 6 - -- F(n) F(n + 1) - 7/22 F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) 22 22 22 3 2 - 5/22 F(n + 1) + 5/22 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 124 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 19 3 37 3 41 2 23 2 2 G(n) = -- F(n) - -- F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 22 22 22 11 115 2 5 39 3 2 19 + --- F(n) F(n + 1) + -- F(n) F(n + 1) + 7/22 F(n) F(n + 1) - -- 11 11 22 6 485 3 4 70 7 145 3 + 25/2 F(n) F(n + 1) - --- F(n) F(n + 1) - -- F(n + 1) + --- F(n + 1) 22 11 22 4 + 7/11 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 125 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 3 4 G(n) = -5/6 F(n) F(n + 1) + 4/21 F(n) + 5/4 F(n) F(n + 1) 2 5 5 2 7 7 - 7/6 F(n) F(n + 1) + 3/4 F(n) F(n + 1) + 1/6 F(n + 1) - 4/21 F(n) 3 - 1/6 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 126 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 16450 3 5 690 2517 690 3897 4 G(n) = ------ F(n) F(n + 1) + ---- + ----- F(n) - ---- F(n + 1) - ----- F(n) 6061 6061 12122 6061 12122 50599 3 24675 7 54075 2 6 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 12122 6061 6061 129475 4 4 9469 2 2 2367 3 + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 12122 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 127 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 49 17 2 1855 3 4 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 638 875 6 1395 2 5 1585 4 3 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 638 319 105 3 20 2 105 2 20 3 - --- F(n) + --- F(n + 1) + --- F(n) - --- F(n + 1) 638 319 638 319 731 2 - --- F(n) F(n + 1) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 128 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1625 8 223 3 3875 7 G(n) = ---- F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 84 28 84 3 5 50 2 6 2 + 300/7 F(n) F(n + 1) - -- F(n) F(n + 1) - 1/12 F(n) + 1/12 21 3 71 2 2 + 1/7 F(n) F(n + 1) + 1/6 F(n) F(n + 1) - -- F(n) F(n + 1) 28 4 - 136/7 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 129 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 16359 4 13 20 3 427 2 G(n) = ----- F(n + 1) + --- + --- F(n + 1) + --- F(n) F(n + 1) 638 319 319 638 1277 2 2 25 2 41 3 + ---- F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 638 58 58 5677 3 1700 2 6 37575 3 5 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 319 638 16425 8 19450 7 13 3 - ----- F(n + 1) + ----- F(n) F(n + 1) - --- F(n) 638 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 130 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 105 2 7 105 3 G(n) = 9/319 F(n) - --- F(n + 1) - 9/319 F(n) + --- F(n + 1) 638 638 318 2 229 2 1811 2 5 + --- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 638 638 1855 5 2 72 6 105 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 638 319 638 391 6 - --- F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 131 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 25 8 2 G(n) = --- F(n + 1) - 1/12 + 1/12 F(n) - 1/6 F(n) F(n + 1) 12 185 2 2 613 3 275 7 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 84 84 84 400 2 6 4 4 3 + --- F(n) F(n + 1) - 150/7 F(n) F(n + 1) + 9/14 F(n) F(n + 1) 21 4 + 13/6 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 132 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1270675 3 6 731300 8 4875 2 7 G(n) = -------- F(n) F(n + 1) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 12122 6061 12122 254329 4 678 2 601675 9 690 2 - ------ F(n) F(n + 1) - ---- F(n) - ------ F(n + 1) - ---- F(n + 1) 12122 6061 12122 6061 678 1988 48573 2 3 183 + ---- F(n) + ---- F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 6061 6061 12122 15025 3 2 599079 5 - ----- F(n) F(n + 1) + ------ F(n + 1) 6061 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 133 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 309 3 4 1700 4 4 7 G(n) = --- F(n) F(n + 1) + ---- F(n) F(n + 1) + 3/638 F(n) 638 319 153 7 1251 2 4705 3 5 - ---- F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 3190 3190 1276 4 3 2007 5 3 147 6 - 3/319 F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 638 638 7355 6 2 186 3 1203 4 751 4 + ---- F(n) F(n + 1) - ---- F(n + 1) - ---- F(n) - ---- F(n + 1) 2552 1595 2552 1276 1851 5 2 1235 3 1191 731 8 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- + ---- F(n + 1) 3190 1276 2552 2552 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 134 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 2 G(n) = -1/84 F(n) (1226 F(n) F(n + 1) - 350 F(n) F(n + 1) 3 4 2 8 + 62 F(n) F(n + 1) - 3263 F(n + 1) - 5 + 30 F(n + 1) + 3250 F(n + 1) 2 2 6 3 5 - 30 F(n) F(n + 1) + 5 F(n) - 675 F(n) F(n + 1) + 7450 F(n) F(n + 1) 7 - 7700 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 135 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(4 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 105 9 570 5 1649 9 543 8 483 262 G(n) = --- F(n + 1) - --- F(n) + ---- F(n) + ---- F(n) - ---- - --- F(n) 638 319 638 2552 2552 319 63 8 795 7 795 4 4 + ---- F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 2552 638 2552 1791 3 5 314 2 7 1425 2 6 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 319 1276 4551 4 5 555 6 2 9265 5 4 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 1276 638 2031 5 3 4465 8 1153 7 2 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 638 319 705 3 6 - --- F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 136 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 2 2 G(n) = -1/6 F(n) (-5 + F(n + 1) - F(n) F(n + 1) + 2 F(n) - 13 F(n) F(n + 1) 3 4 + 13 F(n) F(n + 1) + 3 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 137 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(n + j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 83 2 4 83 3 3 G(n) = -- F(n) F(n + 1) + 7/22 F(n) F(n + 1) - -- F(n) F(n + 1) 22 22 5 2 2 2 - 3/11 F(n) F(n + 1) - 1/11 F(n) + 1/2 F(n + 1) + 7/11 F(n) F(n + 1) 15 2 3 3 6 - -- F(n) F(n + 1) + 1/11 F(n) - 5/22 F(n + 1) - 3/11 F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 138 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(n + j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 3 3 G(n) = -33/2 F(n) F(n + 1) + 41/2 F(n) F(n + 1) + 13/6 F(n) F(n + 1) 4 6 3 - 1/3 F(n + 1) + 23/3 F(n + 1) + 1/3 F(n) F(n + 1) 2 2 2 4 3 + 5/6 F(n) F(n + 1) - 11/2 F(n) F(n + 1) - 3/2 F(n) F(n + 1) + 1/3 2 2 - 23/3 F(n + 1) - 1/3 F(n) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 139 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 485 3 4 15 2 G(n) = 25/2 F(n) F(n + 1) - --- F(n) F(n + 1) - -- F(n) F(n + 1) 22 22 145 3 3 3 + --- F(n + 1) - 4/11 + 4/11 F(n) + 7/22 F(n) F(n + 1) 22 2 23 2 2 115 2 5 - 4/11 F(n) F(n + 1) - -- F(n) F(n + 1) + --- F(n) F(n + 1) 11 11 17 3 70 7 4 + -- F(n) F(n + 1) - -- F(n + 1) + 3/22 F(n + 1) 11 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 140 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 4 4 5 G(n) = 575/6 F(n) F(n + 1) + 11/2 F(n) F(n + 1) - 8/3 F(n + 1) 7 6 2 + 245/6 F(n + 1) - 95 F(n) F(n + 1) - 11/3 F(n) F(n + 1) 2 3 2 5 3 2 + 7/3 F(n) F(n + 1) - 35/3 F(n) F(n + 1) - 47/6 F(n) F(n + 1) 2 3 - 2/3 F(n) + 8/3 F(n + 1) + 29/2 F(n) F(n + 1) - 245/6 F(n + 1) 3 + 2/3 F(n) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 141 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(n + 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 13 3 20 3 3064 3 211 2 G(n) = ---- F(n) + --- F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 319 638 765 2 2 4725 3 5 20 275 7 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- + --- F(n) F(n + 1) 319 638 319 29 13025 2 6 16425 4 4 4 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + 3/29 F(n) 638 638 2 + 2/29 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 142 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(n + 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 4 29 3 45 2 5 G(n) = -4/11 F(n) - 5/22 F(n + 1) + -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 205 3 4 6 15 2 - --- F(n) F(n + 1) - 5/22 F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 27 2 2 2 70 4 3 - -- F(n) F(n + 1) - 2/11 F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 3 3 3 + 5/22 F(n + 1) + 4/11 F(n) + 6/11 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 143 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 245 7 481 3 15 8 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) - -- F(n) + 5/22 44 44 22 745 2 6 267 3 51 3 2 + --- F(n) F(n + 1) + --- F(n) F(n + 1) + -- F(n) F(n + 1) 22 44 11 73 4 4 201 2 2 - -- F(n) F(n + 1) - 5/11 F(n) F(n + 1) - --- F(n) F(n + 1) 22 22 27 2 3 475 7 545 3 5 - -- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 22 44 22 + 5/11 F(n) - 5/22 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 144 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 325 9 7 19 3 3 G(n) = --- F(n) + 1/14 F(n + 1) - -- F(n) - 1/14 F(n + 1) 168 84 6 115 2 3 899 3 2 - 1/7 F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 28 168 425 3 6 317 4 425 4 5 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 24 84 28 3 4 5 41 5 + 2/7 F(n) F(n + 1) + 4/21 F(n + 1) - -- F(n) - 4/21 F(n + 1) 24 425 2 7 4 3 2 5 - --- F(n) F(n + 1) + 1/2 F(n) F(n + 1) - 1/2 F(n) F(n + 1) 84 5 2 - 1/7 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 145 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(n + 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 94 8 20 8 49369 4 16121 8 G(n) = ---- F(n) + --- F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 4785 3190 105 7 283 3 5144 9 20 9 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) - --- F(n + 1) 319 638 957 319 1870 8 20533 2 7 170 6 2 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + --- F(n) F(n + 1) 87 1914 319 374731 6 3 14434 7 2 228 3 5 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + --- F(n) F(n + 1) 9570 957 319 35 7 189 2 2 42573 2 3 + --- F(n) F(n + 1) + --- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 638 3190 260 2 6 5426 - --- F(n) F(n + 1) + ---- F(n) 319 957 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 146 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 2 4 G(n) = 1/6 F(n) (11 F(n) F(n + 1) + 6 F(n) F(n + 1) + F(n) - F(n) 2 4 3 3 2 - 22 F(n) F(n + 1) + 5 F(n) F(n + 1) - 85 F(n) F(n + 1) + 7 F(n + 1) 4 5 2 2 - 3 F(n + 1) + 15 F(n) F(n + 1) - 14 F(n) F(n + 1) 4 2 + 80 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 147 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 925 8 42 5 93 3 21 21 G(n) = ---- F(n + 1) + -- F(n + 1) + -- F(n) F(n + 1) - -- + -- F(n) 22 11 22 22 22 89 575 2 6 89 4 - -- F(n + 1) + --- F(n) F(n + 1) - -- F(n) F(n + 1) 22 22 11 35 2 2 50 2 3 259 3 2 - -- F(n) F(n + 1) - -- F(n) F(n + 1) + --- F(n) F(n + 1) 22 11 22 2425 3 5 285 3 2075 7 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 22 22 22 951 4 + --- F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 148 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 6 6 8 G(n) = -4 F(n) + 1/2 F(n) + 4/5 F(n + 1) + 3/10 F(n + 1) 4 4 6 2 7 - 85/6 F(n) F(n + 1) - 131/6 F(n) F(n + 1) - 55/6 F(n) F(n + 1) 46 5 2 4 + -- F(n) F(n + 1) - 16/3 F(n) F(n + 1) + 7 F(n) F(n + 1) 15 4 86 5 2 - 19/5 F(n + 1) - -- F(n) F(n + 1) - 4/5 F(n + 1) + 7/2 15 629 5 3 116 3 + --- F(n) F(n + 1) + --- F(n) F(n + 1) 15 15 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 149 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n - j) F(2 n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 29 2 4 1425 4 5 G(n) = 9/22 F(n) F(n + 1) - -- F(n) F(n + 1) + ---- F(n) F(n + 1) 22 22 59 3 2 104 4 2 74 5 5 + -- F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) F(n + 1) + 5/2 F(n) 22 11 11 101 3 3 41 4 301 2 3 + --- F(n) F(n + 1) - -- F(n) F(n + 1) + --- F(n) F(n + 1) 22 22 22 2 7 225 3 6 29 2 2 - 25/2 F(n) F(n + 1) + --- F(n) F(n + 1) - -- F(n) - 5/22 F(n + 1) 22 22 1715 5 4 13 9 - ---- F(n) F(n + 1) - -- F(n) + 5/22 F(n + 1) 22 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 150 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n - j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 6 6 G(n) = -1/30 F(n) (-22 F(n + 1) - 45 + 35 F(n) + 47 F(n + 1) 4 4 5 5 3 + 1185 F(n) F(n + 1) - 11 F(n) F(n + 1) + 2730 F(n) F(n + 1) 6 2 7 - 3810 F(n) F(n + 1) - 1125 F(n) F(n + 1) - 119 F(n) F(n + 1) 2 6 3 3 3 - 510 F(n) F(n + 1) + 1530 F(n) F(n + 1) - 470 F(n) F(n + 1) 4 2 8 + 575 F(n) F(n + 1) + 10 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 151 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(2 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 185 185 2 137 2 137 5 G(n) = --- F(n + 1) - --- F(n + 1) - --- F(n) + --- F(n) 638 638 638 638 137 2 3 45 2 4 137 3 2 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 638 72 3 3 137 4 183 4 2 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 14 5 137 4 5 - -- F(n) F(n + 1) + --- F(n) F(n + 1) - 2/319 F(n) F(n + 1) 29 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 152 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(3 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 145 7 23 3 145 3 2 G(n) = --- F(n + 1) - -- F(n) - --- F(n + 1) - 2/7 F(n) F(n + 1) 28 84 28 2 5 345 6 2 23 - 5/7 F(n) F(n + 1) - --- F(n) F(n + 1) + 12/7 F(n) F(n + 1) + -- F(n) 28 84 325 3 4 + --- F(n) F(n + 1) 28 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 153 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(3 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 40 5 340 4 3 317 6 G(n) = --- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 638 20 7 225 3 4 40 5 + --- F(n) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 638 319 1399 5 2 19 6 20 6 225 2 - ---- F(n) F(n + 1) + -- F(n) F(n + 1) - --- F(n) + --- F(n + 1) 638 22 319 638 247 7 20 6 38 31 3 - --- F(n + 1) - --- F(n + 1) + --- F(n) F(n + 1) + --- F(n + 1) 638 319 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 154 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 106 3 5 1626 2 2 921 4 G(n) = --- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 209 6061 12122 921 2 3 921 4 10885 5 3 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 6061 6061 21741 6 2 1525 7 921 3 2 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 12122 6061 12122 216 7 659 8 921 5 615 - ---- F(n) F(n + 1) - ----- F(n) + ----- F(n) + ----- F(n + 1) 6061 12122 12122 12122 353 4 615 + ----- F(n) - ----- 12122 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 155 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 8 19 8 6 4 G(n) = -5/4 F(n + 1) + 5/21 F(n + 1) - -- F(n) - 1/4 F(n) - 5/7 F(n + 1) 84 10 95 4 4 3 3 7 + -- + -- F(n) F(n + 1) - 5/2 F(n) F(n + 1) - 10/7 F(n) F(n + 1) 21 42 6 2 3 5 5 - 30/7 F(n) F(n + 1) - 26/7 F(n) F(n + 1) + 3 F(n) F(n + 1) 5 3 2 + 50/7 F(n) F(n + 1) + 5/4 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 156 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1030 8 177 3 177 4 185 3 G(n) = ----- F(n + 1) + --- F(n) - --- F(n) + --- F(n + 1) 29 638 638 638 301 3 185 2 6 2933 2 2 - --- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 638 83 2 27570 7 441 3 + --- F(n) F(n + 1) + ----- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 29 23790 3 5 15 2 775 4 - ----- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n + 1) 319 638 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 157 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 2 G(n) = ) F(j) F(2 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) G(n) = - 2 2 1/12 F(n) (F(n) - 2 F(n + 1)) (3 F(n) - 2 F(n) F(n + 1) - 3 + 2 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 158 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 185 185 3 111 2 63 3 G(n) = --- - --- F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 638 638 638 319 115 2 35 2 2 117 3 161 3 - --- F(n) F(n + 1) + -- F(n) F(n + 1) - --- F(n) - --- F(n) F(n + 1) 638 58 319 319 49 4 + --- F(n) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 159 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 615 615 9213 441 3 G(n) = ----- - ----- F(n + 1) + ----- F(n) - --- F(n) F(n + 1) 12122 12122 12122 551 1281 2 2 4987 2 3 861 3 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - --- F(n) F(n + 1) 1102 12122 551 18237 3 2 651 4 212 4 16989 5 + ----- F(n) F(n + 1) + ---- F(n) - --- F(n) F(n + 1) - ----- F(n) 12122 1102 319 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 160 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = -1/12 F(n) (-7 + 9 F(n + 1) - 9 F(n) F(n + 1) + 6 F(n) 2 2 3 4 - 21 F(n) F(n + 1) + 21 F(n) F(n + 1) + F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 161 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 185 185 135 3 G(n) = - --- + --- F(n + 1) + 1/319 F(n) - --- F(n) F(n + 1) 638 638 638 821 2 2 925 2 3 101 3 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 638 638 638 296 3 2 258 4 337 4 333 5 - --- F(n) F(n + 1) + --- F(n) + --- F(n) F(n + 1) - --- F(n) 319 319 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 162 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 3 5 G(n) = -4 F(n) F(n + 1) - 7/2 F(n) F(n + 1) - 22 F(n) F(n + 1) 3 3 6 4 2 + 24 F(n) F(n + 1) + 29/3 F(n + 1) - 3/4 F(n + 1) - 3/4 F(n) 2 3 - 29/3 F(n + 1) + 7/2 F(n) F(n + 1) + 1/2 F(n) F(n + 1) 2 2 + 9/4 F(n) F(n + 1) + 3/4 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 163 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1018 4 5 G(n) = ----- F(n) F(n + 1) - 1/58 F(n) F(n + 1) + 3 F(n) F(n + 1) 319 1651 2 3 2 4 349 3 2 - ---- F(n) F(n + 1) - 9/2 F(n) F(n + 1) + --- F(n) F(n + 1) 638 319 3 3 767 4 4 2 565 5 - 4 F(n) F(n + 1) + --- F(n) F(n + 1) + 9 F(n) F(n + 1) - --- F(n) 638 638 185 185 2 565 2 - --- F(n + 1) + --- F(n + 1) + --- F(n) 638 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 164 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 73 2 2 3 2 5 G(n) = --- F(n) F(n + 1) + 25/4 F(n) F(n + 1) - 20/3 F(n) F(n + 1) 12 3 2 6 3 4 - 75/4 F(n) F(n + 1) - 115 F(n) F(n + 1) + 325/3 F(n) F(n + 1) 3 3 233 2 4 + 5/4 F(n) - 145/3 F(n + 1) + --- F(n) F(n + 1) + 25/2 F(n) F(n + 1) 12 5 7 - 25/4 F(n + 1) + 145/3 F(n + 1) - 5/4 F(n) + 25/4 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 165 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(2 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 83 6 3 3 96 5 G(n) = --- F(n + 1) - 5/22 F(n + 1) - 9/22 F(n) + -- F(n) F(n + 1) 22 11 102 3 3 2 2 - --- F(n) F(n + 1) + 3/22 F(n) F(n + 1) - 2/11 F(n) F(n + 1) 11 13 39 2 4 2 2 - -- F(n) F(n + 1) + -- F(n) F(n + 1) + 9/22 F(n) + 4 F(n + 1) 11 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 166 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(2 n - j) F(3 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 96 7 76 2 52 4 3 2711 5 2 G(n) = ---- F(n) - --- F(n + 1) + -- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 29 1595 192 5 20 30 6 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 319 192 5 40 6 301 2 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 1595 96 6 96 6 57 7 157 3 + --- F(n) + --- F(n + 1) + ---- F(n + 1) - ---- F(n + 1) 319 319 1595 1595 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 167 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(2 n - j) F(3 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 415 4 3 80 6 G(n) = 5/22 F(n + 1) + --- F(n) F(n + 1) + -- F(n) F(n + 1) - 5/22 22 11 79 2 4 3 - -- F(n) F(n + 1) + 4/11 F(n) - 9/22 F(n) F(n + 1) 11 175 2 5 3 47 2 - --- F(n) F(n + 1) - 3/22 F(n) + -- F(n) F(n + 1) 11 22 2 2 3 95 3 4 - 5/11 F(n) F(n + 1) - 5/22 F(n) F(n + 1) - -- F(n) F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 168 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(2 n - j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 G(n) = -1/28 F(n + 1) F(n) (-685 F(n + 1) F(n) + 99 F(n + 1) F(n) 6 2 3 3 4 2 + 290 F(n + 1) - 286 F(n + 1) + 670 F(n + 1) F(n) - 70 F(n + 1) F(n) 2 - 18 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 169 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(2 n - j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 42 11268 4 51 2 1755 2 2 G(n) = --- + ----- F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 319 319 25 2 27570 7 4854 3 + --- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 23790 3 5 530 3 185 2 6 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 319 42 3 1030 8 20 3 - --- F(n) - ---- F(n + 1) + --- F(n + 1) 319 29 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 170 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(2 n - j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 8 G(n) = -5/22 F(n + 1) - 6/11 F(n) + 6/11 F(n) + 5/22 F(n + 1) 4 305 2 2 19 2 3 + 1/22 F(n) F(n + 1) + --- F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 19 3 2 90 3 5 37 4 - -- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 22 960 4 4 410 7 975 2 6 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 11 11 11 35 3 413 3 - -- F(n) F(n + 1) - --- F(n) F(n + 1) 11 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 171 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(2 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 3 G(n) = 5/22 - 5/22 F(n + 1) + 3/22 F(n) F(n + 1) + 2/11 F(n) F(n + 1) 2 2 2 3 10 3 - 2/11 F(n) F(n + 1) + F(n) F(n + 1) - 9/22 F(n) - -- F(n) F(n + 1) 11 4 + 2/11 F(n) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 172 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(2 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 20 233 256 3 G(n) = - --- + --- F(n + 1) + --- F(n) - --- F(n) F(n + 1) 319 319 319 319 416 2 2 216 2 3 544 3 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 319 342 3 2 200 4 194 4 413 5 + --- F(n) F(n + 1) + --- F(n) - --- F(n) F(n + 1) - --- F(n) 319 319 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 173 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(2 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 27 27 2 2 51 2 4 G(n) = -- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 14 28 14 3 3 2 - 3/2 F(n) F(n + 1) + 3/14 F(n) F(n + 1) + 9/28 - 9/28 F(n) 179 2 179 6 4 5 - --- F(n + 1) + --- F(n + 1) - 9/28 F(n + 1) - 99/7 F(n) F(n + 1) 28 28 3 3 + 33/2 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 174 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(2 n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 G(n) = - 5/22 + 5/22 F(n + 1) + 3/11 F(n) - 9/22 F(n) F(n + 1) 2 2 18 2 3 3 - 5/11 F(n) F(n + 1) + -- F(n) F(n + 1) - 5/22 F(n) F(n + 1) 11 27 3 2 4 4 5 - -- F(n) F(n + 1) + 4/11 F(n) + 5/11 F(n) F(n + 1) - 9/22 F(n) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 175 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(2 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1293 3 2 1222 3 3 934 4 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 465 4 2 1183 5 327 4 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 319 304 5 670 2 3 72 2 4 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - -- F(n) F(n + 1) 319 319 29 20 20 2 318 2 318 5 - --- F(n + 1) + --- F(n + 1) + --- F(n) - --- F(n) 319 319 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 176 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(2 n - j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 16 203 2 37 5 6 G(n) = 6/11 F(n) + -- F(n + 1) + --- F(n + 1) - -- F(n + 1) - 9 F(n + 1) 11 22 22 75 4 5 2 3 - 6/11 F(n) + -- F(n) F(n + 1) + 21 F(n) F(n + 1) - 1/22 F(n) F(n + 1) 22 2 4 93 3 2 3 3 + 9/2 F(n) F(n + 1) - -- F(n) F(n + 1) - 22 F(n) F(n + 1) 22 29 - -- F(n) F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 177 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(2 n - j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 37 6 56 2 G(n) = --- F(n) F(n + 1) + 35/2 F(n) F(n + 1) + -- F(n) F(n + 1) 11 11 2 5 24 3 3 3 4 - 75/2 F(n) F(n + 1) - -- F(n) F(n + 1) - 10 F(n) F(n + 1) 11 4 3 15 2 375 2 61 5 + 45 F(n) F(n + 1) + -- F(n) - --- F(n) F(n + 1) + -- F(n) F(n + 1) 22 22 22 92 2 4 89 4 2 15 3 3 - -- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) + 5/22 F(n + 1) 11 11 22 2 - 5/22 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 178 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(3 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2389 1070 2389 2195 3 G(n) = ---- F(n) - ---- F(n + 1) - ---- + ---- F(n) F(n + 1) 6061 6061 6061 6061 16185 3 5 44976 3 255 2 6 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 551 6061 1102 15415 8 18755 7 2459 2 2 - ----- F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 1102 551 1102 176483 4 + ------ F(n + 1) 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 179 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 3 j) F(3 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 6 G(n) = 1/84 F(n) (98 F(n) F(n + 1) + 14 F(n + 1) + 54 F(n) F(n + 1) 7 6 2 5 - 26 F(n + 1) - 28 F(n) F(n + 1) + 196 F(n) F(n + 1) 5 2 3 7 - 322 F(n) F(n + 1) + 14 F(n + 1) - 7 F(n) + 7 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 180 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 3 j) F(3 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2715 4 3 263 2 6 2715 6 G(n) = ----- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 319 638 1510 6 2 125 3 543 2 5 20 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- 319 638 638 319 1875 3 159 2 137 3 5 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 1276 638 44 921 7 20 3 109 3 153 8 + ---- F(n) F(n + 1) + --- F(n + 1) - --- F(n) + --- F(n) 1276 319 58 638 9105 5 3 543 7 - ---- F(n) F(n + 1) + --- F(n) 1276 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 181 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 3 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 545 3 3 57 1070 6 G(n) = ----- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n + 1) 24244 116 6061 78042 4 30575 3 6 62946 3 2 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 1102 6061 61575 4 5 1070 9 2249 6 78695 5 + ----- F(n) F(n + 1) + ---- F(n + 1) - ----- F(n) + ----- F(n) 12122 6061 12122 12122 77400 2 7 171660 5 4 32063 4 - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 6061 6061 9105 5 8451 5 38223 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) 24244 24244 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 182 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 3 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 8 3 G(n) = -1/84 F(n) (30 - 5 F(n + 1) + 4125 F(n + 1) + 1779 F(n) F(n + 1) 4 2 2 3 2 - 4152 F(n + 1) - 490 F(n) F(n + 1) + 13 F(n) F(n + 1) - 30 F(n) 2 6 3 5 + 5 F(n) F(n + 1) - 150 F(n) F(n + 1) + 8850 F(n) F(n + 1) 7 - 9975 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 183 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 3 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 265 4 7 3644 4 5 G(n) = --- F(n) - 9/29 F(n) F(n + 1) + ---- F(n) F(n + 1) 638 319 18 4 4 5 3 20 625 2 3 - -- F(n) F(n + 1) - 9/29 F(n) F(n + 1) + --- + --- F(n) F(n + 1) 29 319 22 9869 2 7 697 3 2919 3 2 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 319 3 5 22353 3 6 1261 4 + 9/29 F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 638 23103 5 4 122 8 1093 2 2 - ----- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 638 18 2 6 305 5 20 + -- F(n) F(n + 1) - --- F(n) - --- F(n + 1) 29 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 184 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(3 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 20 2 96 119 73 2 G(n) = ---- F(n + 1) + --- F(n + 1) + --- F(n) - --- F(n) F(n + 1) - --- F(n) 319 319 319 319 638 357 2 3 43 3 2 177 4 + --- F(n) F(n + 1) - -- F(n) F(n + 1) + --- F(n) F(n + 1) 638 58 319 119 5 - --- F(n) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 185 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 3 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = -1/84 F(n) (-39 + 41 F(n + 1) - 41 F(n) F(n + 1) + 39 F(n) 2 2 3 - 125 F(n) F(n + 1) + 125 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 186 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 3 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1070 1070 702 1773 3 G(n) = - ---- + ---- F(n + 1) + ---- F(n) - ---- F(n) F(n + 1) 6061 6061 6061 6061 8557 2 2 858 2 3 2621 3 - ----- F(n) F(n + 1) + --- F(n) F(n + 1) - ----- F(n) F(n + 1) 12122 551 12122 1199 3 2 3514 4 33 4 286 5 - ---- F(n) F(n + 1) + ---- F(n) + -- F(n) F(n + 1) - --- F(n) 1102 6061 58 551 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 187 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2831 3 3 2573 5 2199 2 23 3 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n + 1) - --- F(n) 319 319 638 319 2239 6 23 2 357 2 13 2 - ---- F(n + 1) + --- F(n) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 638 319 638 638 357 1217 2 4 20 3 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n + 1) 638 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 188 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 5 G(n) = 1/84 F(n) (-55 F(n) + 55 F(n) - 50 F(n + 1) + 48 F(n + 1) 2 4 2 3 + 75 F(n) F(n + 1) - 65 F(n) F(n + 1) + 108 F(n) F(n + 1) 3 2 4 2 - 262 F(n) F(n + 1) + 281 F(n) F(n + 1) - 135 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 189 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 8651 5 39077 6 1383 2 1070 5 G(n) = ------ F(n) + ----- F(n) - ---- F(n) - ---- F(n + 1) 12122 12122 551 6061 1070 6 929 4 2000 5 + ---- F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 6061 6061 7542 2 3 5013 2 4 2513 3 2 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 1102 6061 87453 3 3 5950 4 21331 5 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 12122 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 190 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 18 3 537 2 149 2 2 G(n) = --- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 29 319 319 248 3 95 3 4 80 6 - --- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 319 22 11 175 2 5 415 4 3 20 3 67 4 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n + 1) - --- F(n) 11 22 319 638 20 27 3 2105 2 + --- + --- F(n) - ---- F(n) F(n + 1) 319 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 191 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 G(n) = -1/84 F(n) (-247 F(n) F(n + 1) - 3500 F(n) F(n + 1) 2 4 3 3 4 - 980 F(n) F(n + 1) + 4060 F(n) F(n + 1) + 59 F(n + 1) 6 3 + 1540 F(n + 1) + 477 F(n) F(n + 1) - 118 F(n) F(n + 1) 2 2 2 4 2 + 306 F(n) F(n + 1) - 94 F(n) + 94 F(n) - 1597 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 192 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 255 6 2 469 3 3815 5 3 G(n) = - --- + --- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 22 638 319 921 3 151 2 6 106 7 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 638 638 319 164 2 3 1983 3 2 91 3 5 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) + -- F(n) F(n + 1) 319 638 29 1319 4 515 20 59 8 248 5 + ---- F(n) F(n + 1) - --- F(n) + --- F(n + 1) + --- F(n) + --- F(n) 638 638 319 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 193 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(3 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 23790 3 5 27570 7 4709 3 G(n) = ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 153 2 1697 2 2 185 2 6 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 319 15 2 71 71 3 185 3 + --- F(n) F(n + 1) + --- - --- F(n) + --- F(n + 1) 638 319 319 638 443 3 1030 8 22333 4 - --- F(n) F(n + 1) - ---- F(n + 1) + ----- F(n + 1) 319 29 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 194 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 2 j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3239 8 2613 4 12345 3 6 G(n) = ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 12122 6061 12122 26963 6 3 2154 2 3 46643 5 4 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 6061 12122 1521 1698 3 3 4773 2 4 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 6061 12122 57 5 4773 4 2 615 9 - --- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n + 1) 319 12122 12122 3063 9 849 6 20137 7 2 615 6 - ----- F(n) - ---- F(n) - ----- F(n) F(n + 1) + ----- F(n + 1) 12122 6061 12122 12122 4761 + ----- F(n) 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 195 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 2 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 2 2 G(n) = 1/84 F(n) (33 F(n + 1) - 1725 F(n) F(n + 1) + 447 F(n) F(n + 1) 3 4 7 + 3 F(n) F(n + 1) + 43 F(n) - 33 F(n) F(n + 1) + 1725 F(n) F(n + 1) 2 6 3 5 4 4 - 3975 F(n) F(n + 1) - 600 F(n) F(n + 1) + 4125 F(n) F(n + 1) - 31 2 - 12 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 196 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 2 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 83 5 83 4 185 9 1699 4 G(n) = --- F(n) - --- F(n) - --- F(n + 1) + ---- F(n) F(n + 1) 319 638 638 638 465 7 3180 7 2 1755 3 5 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 319 1276 970 3 6 465 4 4 135 4 5 + --- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 1755 5 3 1905 5 4 765 6 2 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 1276 319 638 5605 6 3 817 4 483 3 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 638 1276 15 7 185 134 + --- F(n) F(n + 1) + --- - --- F(n) 116 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 197 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(3 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = -1/12 F(n) (-5 + 7 F(n + 1) - 7 F(n) F(n + 1) + 5 F(n) 2 2 3 - 25 F(n) F(n + 1) + 25 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 198 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 2 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 185 185 65 3 G(n) = - --- + --- F(n + 1) + --- F(n) - 9/29 F(n) F(n + 1) 638 638 638 24 2 2 861 2 3 3 - -- F(n) F(n + 1) + --- F(n) F(n + 1) - 3/58 F(n) F(n + 1) 29 638 252 3 2 37 4 217 4 287 5 - --- F(n) F(n + 1) + -- F(n) + --- F(n) F(n + 1) - --- F(n) 319 58 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 199 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 543 5 24177 2 3 34709 2 4 G(n) = --- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 551 12122 12122 47163 3 2 27116 3 3 34959 4 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 12122 6061 12122 8959 4 2 2575 5 12435 4 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 638 12122 615 615 2 6102 2 6102 5 + ----- F(n + 1) - ----- F(n + 1) + ---- F(n) - ---- F(n) 12122 12122 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 200 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 2 G(n) = 1/12 F(n) (-10 F(n + 1) + 8 F(n + 1) + 15 F(n) F(n + 1) 4 2 2 3 - 13 F(n) F(n + 1) - 19 F(n) F(n + 1) + 20 F(n) F(n + 1) 3 2 4 3 - 38 F(n) F(n + 1) + 37 F(n) F(n + 1) - 7 F(n) + 7 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 201 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 37 4 59 5 1183 2 3 G(n) = ---- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 638 997 2 4 70 3 2 427 3 3 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 638 853 4 900 4 2 2047 5 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 319 638 185 185 2 202 2 202 5 - --- F(n + 1) + --- F(n + 1) + --- F(n) - --- F(n) 638 638 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 202 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 3 2 G(n) = -1/12 F(n) (-140 F(n) F(n + 1) - 35 F(n) F(n + 1) - 12 F(n) 4 3 2 2 + 13 F(n + 1) + 71 F(n) F(n + 1) - 26 F(n) F(n + 1) + 48 F(n) F(n + 1) 4 3 3 6 5 + 12 F(n) + 580 F(n) F(n + 1) + 220 F(n + 1) - 500 F(n) F(n + 1) 2 - 231 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 203 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 2 5 3 4 313 3 G(n) = -45 F(n + 1) + 15/2 F(n) F(n + 1) - 100 F(n) F(n + 1) - --- F(n) 319 313 2 3521 2 1583 5 1361 3 3 + --- F(n) + ---- F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 58 58 2909 6 3609 2 - ---- F(n) F(n + 1) + 215/2 F(n) F(n + 1) + ---- F(n) F(n + 1) 638 638 62 2 4 657 6 28895 3 - -- F(n) F(n + 1) - --- F(n + 1) + ----- F(n + 1) 29 58 638 5674 2 - ---- F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 204 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 127 5 2 715 4 2 G(n) = --- F(n) F(n + 1) - 33/4 F(n) - --- F(n + 1) + 2/3 F(n + 1) 12 216 3577 5 3 821 2 6 6 6 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) - 2/3 F(n + 1) + 59/6 F(n) 108 72 433 47 8 1981 6 2 83 7 + --- + -- F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 216 36 108 108 277 7 2 4 3 3 + --- F(n) F(n + 1) + 49/4 F(n) F(n + 1) - 26 F(n) F(n + 1) 108 775 8 - --- F(n) 216 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 205 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(3 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 269 4 4 225 7 25 4 G(n) = --- F(n) F(n + 1) - --- F(n) F(n + 1) - -- F(n) F(n + 1) 22 22 44 3 2 324 3 5 21 4 - 7/4 F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) F(n + 1) 55 22 3 533 3 7 + 7/22 F(n) F(n + 1) + --- F(n) F(n + 1) + 7/22 F(n) F(n + 1) + 5/22 110 17 5 38 4 151 8 + -- F(n) + -- F(n) - --- F(n) + 3/44 F(n) - 5/22 F(n + 1) 44 55 110 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 206 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 7 G(n) = -1/84 F(n) (-2 F(n) F(n + 1) + 1709 F(n) F(n + 1) - 9975 F(n) F(n + 1) 2 2 2 3 5 - 490 F(n) F(n + 1) + 2 F(n + 1) + 8850 F(n) F(n + 1) 8 2 4 3 + 4125 F(n + 1) + 2 - 2 F(n) - 4117 F(n + 1) + 48 F(n) F(n + 1) 2 6 - 150 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 207 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3147 3 5 274 3 6 1389 4 G(n) = ----- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 1276 319 319 291 4 4 1982 4 5 3147 5 3 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 1276 4215 5 4 69 6 2 291 7 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 638 638 293 8 4 69 20 85 4 + --- F(n) F(n + 1) + 5/638 F(n) F(n + 1) + --- F(n) + --- - --- F(n) 29 319 319 319 387 3 513 7 659 9 20 9 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) - --- F(n + 1) 1276 1276 319 319 663 5 - --- F(n) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 208 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 27 6 3 6 4 2 G(n) = --- F(n) - 1/11 F(n) F(n + 1) + 13/2 F(n) F(n + 1) 11 131 4 47 2 2 + --- F(n) F(n + 1) - 2/11 F(n) + -- F(n) - 5/22 F(n + 1) 22 22 29 2 3 615 7 2 675 6 3 - -- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 22 22 22 9 9 36 2 4 29 8 + 1/2 F(n) + 5/22 F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 11 22 97 4 5 46 5 13 - -- F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 209 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(3 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = - 5/22 + 5/22 F(n + 1) - 3/22 F(n) - 3/2 F(n) F(n + 1) 21 2 3 19 3 2 4 12 4 + -- F(n) F(n + 1) - -- F(n) F(n + 1) + F(n) + -- F(n) F(n + 1) 11 11 11 5 - 7/11 F(n) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 210 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1668 6043 2 1688 5 207 6 G(n) = ---- F(n + 1) + ---- F(n + 1) - ---- F(n + 1) - --- F(n + 1) 319 638 319 22 623 623 2 1704 2 3 29 2 4 - --- F(n) + --- F(n) + ---- F(n) F(n + 1) - -- F(n) F(n + 1) 638 638 319 22 5064 3 2 203 3 3 1308 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 11 319 3392 4 523 5 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 319 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 211 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 575 3 4 355 5 2 340 6 19 G(n) = --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) - -- F(n) 42 28 21 42 235 7 19 3 295 2 235 3 + --- F(n + 1) + -- F(n) - --- F(n) F(n + 1) - --- F(n + 1) 28 42 84 28 75 3 2 75 4 2 5 - -- F(n) F(n + 1) + -- F(n) F(n + 1) - 20/3 F(n) F(n + 1) 28 28 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 212 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 5 20 2 3 G(n) = -1/11 F(n) F(n + 1) - 1/22 F(n) F(n + 1) - -- F(n) F(n + 1) 11 43 2 4 3 2 19 3 3 + -- F(n) F(n + 1) + 1/11 F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 29 4 65 4 2 29 5 13 5 + -- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) 22 22 11 22 13 2 2 + -- F(n) - 5/22 F(n + 1) + 5/22 F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 213 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 3 9167 6 7913 1471 2 G(n) = --- F(n + 1) + ---- F(n) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 1595 3190 638 5789 5 559 2 1079 6 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 1595 319 319 3871 2 4 3475 2 5 4507 3 3 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 319 75 5 2 1868 6 545 7 20 2 + -- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) - --- F(n + 1) 22 319 319 319 6442 2 - ---- F(n) 1595 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 214 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 2 4 15 2 2 G(n) = 9/2 F(n) F(n + 1) - 13 F(n) F(n + 1) + -- F(n) - 5/22 F(n + 1) 11 4 2 6 62 2 + 11 F(n) F(n + 1) + 35/2 F(n) F(n + 1) + -- F(n) F(n + 1) 11 2 5 192 2 3 3 - 75/2 F(n) F(n + 1) - --- F(n) F(n + 1) - F(n) F(n + 1) 11 3 4 4 3 103 15 3 - 10 F(n) F(n + 1) + 45 F(n) F(n + 1) - --- F(n) F(n + 1) - -- F(n) 22 11 3 + 5/22 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 215 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 n - j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 43 7 67 4 3 965 4 4 G(n) = --- F(n) - -- F(n) F(n + 1) + --- F(n) F(n + 1) 22 11 44 5 3 57 6 1835 6 2 - 49/8 F(n) F(n + 1) + -- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 176 1075 7 26 6 3 - ---- F(n) F(n + 1) + -- F(n) F(n + 1) + 7/5 F(n + 1) 88 11 179 5 2 171 8 179 7 185 4 - --- F(n) F(n + 1) + --- F(n + 1) - --- F(n + 1) + --- F(n) 55 176 110 176 145 4 525 3 5 34 2 159 - --- F(n + 1) - --- F(n) F(n + 1) + -- F(n) F(n + 1) + --- 88 44 55 176 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 216 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 532197 1252808 4 6 474093 3 2 G(n) = ------- F(n + 1) - ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 4824556 1206139 2412278 377741 2 8 474093 2 7 + ------- F(n) F(n + 1) + ------- F(n) F(n + 1) 1206139 4824556 1422279 2 3 3419649 9 - ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 2412278 1206139 237109 8 2 4266837 8 + ------ F(n) F(n + 1) + ------- F(n) F(n + 1) 109649 4824556 2370465 7 2 9430937 6 4 + ------- F(n) F(n + 1) + ------- F(n) F(n + 1) 2412278 2412278 474093 6 3 948186 5 4 5967 6 - ------- F(n) F(n + 1) - ------- F(n) F(n + 1) + ------- F(n + 1) 1206139 1206139 2412278 254556 2 474093 5 982176 6 474093 9 - ------- F(n + 1) + ------- F(n + 1) + ------- F(n) + ------- F(n) 1206139 4824556 1206139 2412278 2438445 10 204093 5 5 1567857 2 4 - ------- F(n) - ------ F(n) F(n + 1) - ------- F(n) F(n + 1) 2412278 109649 2412278 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 217 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 4 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 108265 4 6 8523 2 4 2619 3 G(n) = ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) 12122 30305 12122 24 2 5 48 3 4 2619 10 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) 6061 6061 12122 3747 4545 2 4428 2 8 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 12122 30305 155476 3 3 19654 3 7 24 4 3 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 30305 6061 6061 48 5 2 24 6 367586 7 3 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 6061 6061 30305 1006 8 2 13440 9 615 10 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n + 1) 319 6061 12122 615 3 + ----- F(n + 1) 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 218 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 4 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 4 5 7 G(n) = 1/420 F(n) (-2 F(n) + 8480 F(n) F(n + 1) - 20 F(n + 1) 5 2 3 2 - 90 F(n + 1) - 1290 F(n) F(n + 1) - 6 F(n) F(n + 1) 4 3 3 6 3 4 - 100 F(n) F(n + 1) - 5035 F(n) F(n + 1) - 40 F(n) F(n + 1) 3 2 2 7 2 5 + 4175 F(n) F(n + 1) + 520 F(n) F(n + 1) + 116 F(n) F(n + 1) 7 2 2 7 9 - 7360 F(n) F(n + 1) + 70 F(n) F(n + 1) - 8 F(n) + 10 F(n) 9 8 + 50 F(n + 1) + 530 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 219 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 4 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 72 6 23457 4 2 5367 2 4 G(n) = ---- F(n) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 12122 12122 5223 3 3 2334 5 3552 5 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 6061 6061 51 2 5157 2 615 2 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n + 1) 12122 12122 12122 615 3 72 3 - ----- F(n + 1) - ---- F(n) 12122 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 220 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 4 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 3 G(n) = 1/924 F(n) (589 F(n) + 756 F(n) - 1345 F(n) - 712 F(n + 1) 5 2 2 + 538 F(n + 1) + 1068 F(n) F(n + 1) - 1534 F(n) F(n + 1) 2 3 3 2 4 - 130 F(n) F(n + 1) - 3840 F(n) F(n + 1) + 4610 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 221 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 4 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5498679 10824030 2 3000912 5 G(n) = ------- F(n + 1) + -------- F(n + 1) - ------- F(n + 1) 2412278 1206139 1206139 66285 6 846027 846027 2 3540258 - ----- F(n + 1) - ------- F(n) + ------- F(n) - ------- F(n) F(n + 1) 7562 1206139 1206139 1206139 13227531 4 77355 5 1430793 2 3 + -------- F(n) F(n + 1) + ----- F(n) F(n + 1) - ------- F(n) F(n + 1) 2412278 3781 2412278 16605 2 4 6492351 3 2 81000 3 3 + ----- F(n) F(n + 1) - ------- F(n) F(n + 1) - ----- F(n) F(n + 1) 3781 1206139 3781 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 222 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2823 3 8505 3 1563 2 2 G(n) = ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 12122 12122 12122 1831 3 4 16147 4 3 103111 5 2 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 24244 6061 24244 545 6 1209 6 12363 2 + --- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 24244 24244 633 7 615 3 4 615 + ----- F(n) + ----- F(n + 1) - 9/6061 F(n) - ----- 12122 12122 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 223 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 4 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 2 4 G(n) = 1/924 F(n) (989 F(n) - 989 F(n) + 705 F(n + 1) - 879 F(n + 1) 5 3 - 5265 F(n) F(n + 1) + 4560 F(n) F(n + 1) + 2857 F(n) F(n + 1) 3 2 2 4 2 + 1758 F(n) F(n + 1) - 3736 F(n) F(n + 1) + 16820 F(n) F(n + 1) 3 3 2 4 - 10840 F(n) F(n + 1) - 5980 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 224 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 4 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5067 3 2439 2 6 33531 3 2 G(n) = ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 12122 24244 276665 5 3 19611 7 71701 3 5 - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 24244 24244 24244 552 8 255 6 2 38117 3 - ---- F(n) + --- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 22 24244 4251 4 15951 4 5295 615 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) - ----- F(n + 1) 6061 24244 24244 12122 6273 5 615 + ----- F(n) + ----- 24244 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 225 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 9414 660481 5 359675 9 G(n) = ----- F(n) F(n + 1) - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 6061 6061 551 1701 2 783931 2 4 282600 3 7 - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 12122 12122 551 1534789 6 5095 3 1070 3 5095 2 + ------- F(n + 1) - ----- F(n) - ---- F(n + 1) + ----- F(n) 6061 12122 6061 12122 10167 2 28900 2 8 254658 3 3 + ----- F(n + 1) - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 1102 551 6061 3513 2 289025 10 + ---- F(n) F(n + 1) - ------ F(n + 1) 6061 1102 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 226 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 3 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 28675 2 8 3 11185 10 G(n) = ------ F(n) F(n + 1) + 4/7 F(n) F(n + 1) - ----- F(n + 1) 364 364 357 2 2012 6 3 8975 9 + --- F(n) - ---- F(n + 1) + 1/14 F(n) F(n + 1) + ---- F(n) F(n + 1) 52 39 91 15 2 2 21551 2 4 2369 - -- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 28 364 91 18789 3 3 27889 5 4 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + 3/28 F(n + 1) 91 182 615 10 89891 2 - --- F(n) + ----- F(n + 1) - 3/28 91 1092 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 227 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 3 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 34 23691 2 5085 2 23651 6 G(n) = --- F(n + 1) - ----- F(n + 1) - ---- F(n) + ----- F(n + 1) 319 638 638 638 23600 9 74647 2 4 136900 3 7 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 319 638 319 151 4 15056 4 2 4157 5 - --- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 319 319 127450 5 5 1123 2 3 795 3 2 - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 319 638 638 42475 4 6 14 5 23 5 5039 6 - ----- F(n) F(n + 1) - --- F(n + 1) + --- F(n) + ---- F(n) 319 319 319 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 228 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 3 358 1635 2 4 G(n) = --- F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 638 210 2 21 2 4050 5 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 638 319 4310 3 3 3505 6 61 2 61 3 - ---- F(n) F(n + 1) - ---- F(n + 1) - --- F(n) + --- F(n) 319 638 638 638 315 2 + --- F(n + 1) 58 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 229 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 3 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 3 G(n) = -1/84 F(n) (-155 F(n) - 54 F(n) + 209 F(n) + 60 F(n + 1) 5 2 4 - 58 F(n + 1) - 90 F(n) F(n + 1) + 300 F(n) F(n + 1) 2 2 3 4 + 138 F(n) F(n + 1) - 510 F(n) F(n + 1) + 160 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 230 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 3 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2226 21 4 15014 5 G(n) = ----- F(n) F(n + 1) - --- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 418 6061 201 4 2640 4 2 8581 5 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 418 551 6061 35693 1336 5 8359 6 11949 7647 2 + ----- F(n) - ---- F(n) + ----- F(n) + ----- F(n + 1) - ---- F(n + 1) 60610 1045 12122 60610 6061 71 5 8717 6 - --- F(n + 1) + ---- F(n + 1) 190 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 231 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 3 7 69 6 20 G(n) = ---- F(n + 1) + 7/319 F(n) + ---- F(n) F(n + 1) + --- 319 3190 319 459 3 887 2 97 2 2 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 638 3190 319 233 3 2197 4 3 3005 5 2 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 638 638 1689 6 1198 2 27 4 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) 1595 1595 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 232 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 2 G(n) = -1/84 F(n) (-1637 F(n + 1) + 87 F(n) - 87 F(n) + 502 F(n) F(n + 1) 3 2 2 2 4 - 138 F(n) F(n + 1) + 333 F(n) F(n + 1) - 965 F(n) F(n + 1) 3 4 6 5 - 264 F(n) F(n + 1) + 69 F(n + 1) + 1570 F(n + 1) - 3575 F(n) F(n + 1) 3 3 + 4105 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 233 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 632 999 6 15437 2 G(n) = ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 30305 12122 6061 799 2 4 59935 4 3 8694 5 - --- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 418 6061 6061 59350 5 2 10657 6 4017 2 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 12122 12122 6553 5 1070 7 13295 7 734 2 + ----- F(n) F(n + 1) + ---- F(n + 1) - ----- F(n) - ---- F(n) 30305 6061 12122 1595 94367 6 1070 6 + ----- F(n) - ---- F(n + 1) 60610 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 234 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 35 4 95 3 5 20 20 75 G(n) = ---- F(n) - -- F(n) F(n + 1) - --- + --- F(n + 1) + --- F(n) 638 11 319 319 638 653 4 17 4 8227 2 2 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 638 638 318 2 3 747 3 925 3 2 + --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 638 415 7 1915 4 4 11769 3 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 11 22 319 1945 2 6 - ---- F(n) F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 235 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 159 4 2 4 4 3 3 G(n) = --- F(n) F(n + 1) - 137/7 F(n) F(n + 1) - 17/2 F(n) F(n + 1) 14 533 6 2 7 283 3 5 - --- F(n) F(n + 1) + 86/7 F(n) F(n + 1) + --- F(n) F(n + 1) 42 21 188 5 3 19 4 65 4 19 19 8 + --- F(n) F(n + 1) + -- F(n + 1) - -- F(n) - -- - -- F(n + 1) 21 14 84 21 42 51 5 13 6 47 6 39 - -- F(n) F(n + 1) + --- F(n + 1) + -- F(n) - -- F(n) F(n + 1) 10 140 28 35 13 2 - --- F(n + 1) 140 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 236 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 9157 4 5 1451 8 911 G(n) = ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n + 1) 638 1914 957 777 2 2571 2 362 3 3 2749 6 + ---- F(n + 1) + ---- F(n) - --- F(n) F(n + 1) - ---- F(n) 1276 1276 319 1276 851 5 869 5 1661 5 4 2795 5 + --- F(n + 1) - --- F(n) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 957 174 174 638 697 6 2586 3 2 9421 8 - ---- F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 1276 319 638 15903 7 2 1123 4 2 4913 9 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) 638 319 957 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 237 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 74287 6 13604 2 3 3195967 10 G(n) = ----- F(n + 1) + ----- F(n) F(n + 1) + ------- F(n) 8525 2871 494450 1022873 10 2429 1945 1714574 2 - ------- F(n + 1) + ---- F(n + 1) + ---- F(n) - ------- F(n + 1) 494450 2871 1914 247225 1627046 2 10535 8 553 5 860 9 - ------- F(n) - ----- F(n) F(n + 1) + --- F(n + 1) - --- F(n) 247225 2871 638 957 4085 9 1129076 9 2233316 7 3 - ---- F(n + 1) + ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 2871 49445 49445 19809 8 2 21715 7 2 3993 9 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 5742 899 3384454 5 121 4 15265 8 - ------- F(n) F(n + 1) + --- F(n) F(n + 1) + ----- F(n) F(n + 1) 247225 174 5742 26896 - ------ F(n) F(n + 1) 247225 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 238 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 2 G(n) = -1/12 F(n) (8 F(n + 1) - 6 F(n + 1) - 12 F(n) F(n + 1) 4 2 2 3 + 30 F(n) F(n + 1) + 20 F(n) F(n + 1) - 70 F(n) F(n + 1) 3 2 3 5 + 30 F(n) F(n + 1) - 8 F(n) + 23 F(n) - 15 F(n)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 239 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 2 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5565 5 2 3 2415 2 4 G(n) = ---- F(n) F(n + 1) - 1/2 F(n) F(n + 1) + ---- F(n) F(n + 1) 319 638 3 2 5810 3 3 73 - 11/2 F(n) F(n + 1) - ---- F(n) F(n + 1) - -- F(n) F(n + 1) 319 29 4 5 2380 6 228 + 11/2 F(n) F(n + 1) - 5/2 F(n + 1) - ---- F(n + 1) - --- F(n) 319 319 228 2 705 4945 2 + --- F(n) + --- F(n + 1) + ---- F(n + 1) 319 319 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 240 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 17631 6 615 3 9990 7 37611 2 G(n) = ------ F(n) - ----- F(n + 1) - ---- F(n) + ----- F(n) 12122 12122 6061 12122 615 2 7584 5 14679 6 + ----- F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 12122 6061 12122 10590 3 3 12915 4 2 6135 4 3 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 1102 1102 1865 5 2 4517 6 34919 2 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 551 638 12122 295 3 4 72570 5 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 12122 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 241 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 4 6 G(n) = 73/4 F(n) F(n + 1) + 425/4 F(n) F(n + 1) - 335/3 F(n) F(n + 1) 13 3 565 7 2 4 + -- F(n) + --- F(n + 1) - 16/3 F(n) F(n + 1) + 21/2 F(n) F(n + 1) 12 12 3 2 2 5 2 3 - 46/3 F(n) F(n + 1) - 15/2 F(n) F(n + 1) + 29/6 F(n) F(n + 1) 5 13 565 3 - 31/6 F(n + 1) - -- F(n) + 31/6 F(n + 1) - --- F(n + 1) 12 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 242 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 481 3 735 7 1887 7 2187 2 G(n) = ----- F(n + 1) - --- F(n) + ---- F(n + 1) - ---- F(n) F(n + 1) 1595 638 3190 1595 1422 3 4 10639 5 3569 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 3190 3190 446 2 1121 6 6378 5 2 + ---- F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 1595 319 1595 5955 4 3 1817 6 4 2 + ---- F(n) F(n + 1) - ---- F(n + 1) - 2 F(n) F(n + 1) 638 3190 763 3 3 735 6 + --- F(n) F(n + 1) + --- F(n) 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 243 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 7 5 G(n) = -1/60 F(n) (-24 F(n) + 105 F(n) - 81 F(n) + 368 F(n + 1) 3 2 4 3 + 28 F(n + 1) + 266 F(n) F(n + 1) - 1610 F(n) F(n + 1) 3 4 4 5 2 + 1085 F(n) F(n + 1) + 830 F(n) F(n + 1) + 539 F(n) F(n + 1) 6 7 4 - 430 F(n) F(n + 1) - 298 F(n + 1) - 88 F(n + 1) - 690 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 244 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7386 5 3 1975 7 3341 2 G(n) = ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 3190 185 7 1191 7 13723 6 2 - --- F(n + 1) - ---- F(n) + ----- F(n) F(n + 1) 638 638 1595 10734 6 12739 2 2 634 3 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 1595 3190 1595 3419 2 901 7 468 6 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 1595 1595 1595 1999 4 3 185 5 2 185 8 1191 8 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n + 1) + ---- F(n) 319 319 638 638 6130 4 4 + ---- F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 245 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 6 G(n) = -1/12 F(n) (-17 F(n) F(n + 1) - 34 - 127 F(n + 1) + 34 F(n) 6 7 4 4 + 144 F(n + 1) - 1500 F(n) F(n + 1) - 200 F(n) F(n + 1) 2 2 3 4 2 - 870 F(n) F(n + 1) + 289 F(n) F(n + 1) + 410 F(n) F(n + 1) 5 5 3 4 - 122 F(n) F(n + 1) + 2280 F(n) F(n + 1) - 721 F(n + 1) 5 3 8 - 288 F(n) F(n + 1) - 18 F(n) F(n + 1) + 740 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 246 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 10 63 G(n) = -5/22 F(n) - 5/22 F(n + 1) + 5/22 F(n + 1) - -- F(n) F(n + 1) 11 8708 5 40 6 8770 9 - ---- F(n) F(n + 1) + -- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 11 11 2 7709 2 4 90 2 5 + 3/22 F(n) F(n + 1) + ---- F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 21540 2 8 1596 3 3 255 3 4 - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 11 11 22 4 2 375 4 3 35905 4 6 + 51 F(n) F(n + 1) + --- F(n) F(n + 1) + ----- F(n) F(n + 1) 22 22 2 805 3 7 95 2 + 5/22 F(n) + --- F(n) F(n + 1) - -- F(n) F(n + 1) 11 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 247 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 13 5 13 6 G(n) = -5/22 F(n + 1) + 5/22 F(n + 1) - -- F(n) + -- F(n) 11 11 69 2 3 43 2 4 13 3 2 - -- F(n) F(n + 1) + -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 11 41 3 3 18 4 32 4 2 - -- F(n) F(n + 1) + -- F(n) F(n + 1) + -- F(n) F(n + 1) 11 11 11 29 5 4 5 - -- F(n) F(n + 1) + 7/22 F(n) F(n + 1) - 5/11 F(n) F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 248 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 12635 3 983 3 435 7 9780 2 983 2 G(n) = ----- F(n + 1) - --- F(n) - --- F(n + 1) + ---- F(n + 1) + --- F(n) 319 638 11 319 638 22320 5 24320 3 3 12281 2 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 638 3055 2185 6 2403 2 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 22 319 4080 2 4 195 2 5 1665 3 4 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 22 22 9800 6 - ---- F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 249 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 335 5 50 2 465 8 G(n) = 5/21 F(n + 1) - --- F(n) F(n + 1) + -- F(n) + --- F(n) 42 21 28 13 8 13 4 352 4 13 145 3 5 - -- F(n + 1) + -- F(n + 1) - --- F(n) - -- + --- F(n) F(n + 1) 42 14 21 21 14 1115 6 2 850 7 155 4 2 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 42 21 21 327 5 3 20 2 4 45 6 6 - --- F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) - 5/21 F(n + 1) 14 21 28 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 250 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 79 2 25 6 5 G(n) = -- F(n) F(n + 1) + -- F(n) F(n + 1) - 23/5 F(n) F(n + 1) 22 22 61 3 7 2 173 2 + -- F(n + 1) - 8/11 F(n) + 8/11 F(n) + --- F(n + 1) 22 110 18 116 2 5 2 4 + -- F(n) F(n + 1) + --- F(n) F(n + 1) - 2 F(n) F(n + 1) 55 11 41 2 5 357 5 2 + -- F(n) F(n + 1) + 7/2 F(n) F(n + 1) - --- F(n) F(n + 1) 11 22 6 28 7 - 9/5 F(n + 1) - -- F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 251 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 109295 6 103210 3 4 863 3 G(n) = ------ F(n) F(n + 1) - ------ F(n) F(n + 1) - --- F(n) 319 319 319 45940 3 8953 4 11767 17781 2 + ----- F(n + 1) + ---- F(n + 1) - ----- - ----- F(n) F(n + 1) 319 1276 2204 319 675 8 7865 2 2 6550 2 5 - --- F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 418 1102 319 4375 2 6 79505 3 79554 3 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 836 6061 6061 2875 7 945 2 6725 8 45960 7 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) - ----- F(n + 1) 418 58 836 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 252 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 51 3 7 133 3 4 G(n) = --- F(n) F(n + 1) + 5/2 F(n) F(n + 1) + --- F(n) F(n + 1) + 5/22 22 22 49 2 37 8 65 7 68 4 3 + -- F(n) F(n + 1) + -- F(n) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 22 22 22 11 225 5 3 6 3 - --- F(n) F(n + 1) + 9/2 F(n) F(n + 1) - 21/2 F(n) F(n + 1) 11 32 3 5 6 2 133 5 2 - -- F(n) F(n + 1) + 65/2 F(n) F(n + 1) - --- F(n) F(n + 1) 11 22 23 6 21 7 3 - -- F(n) F(n + 1) - -- F(n) - 5/22 F(n + 1) 11 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 253 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 n - j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 9776 5 9 8 7 G(n) = ---- F(n + 1) - 1825/2 F(n + 1) - 200 F(n + 1) + 500 F(n) F(n + 1) 11 2083 3 3 5 461 3 - ---- F(n) F(n + 1) - 775/2 F(n) F(n + 1) - --- F(n) F(n + 1) 22 22 2 6 29 29 4337 4 - 50 F(n) F(n + 1) - -- F(n) + 24 F(n + 1) + -- + ---- F(n + 1) 11 11 22 1043 2 2 4391 2 3 2 7 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - 225/2 F(n) F(n + 1) 22 22 2567 3 2 3 6 4120 4 - ---- F(n) F(n + 1) - 3675/2 F(n) F(n + 1) - ---- F(n) F(n + 1) 22 11 8 + 2250 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 254 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(2 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 21 2 G(n) = 5/22 F(n + 1) - 5/22 F(n + 1) - 9/11 F(n) F(n + 1) + -- F(n) F(n + 1) 22 2 2 3 + 7/11 F(n) - 3/22 F(n) F(n + 1) - 7/11 F(n) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 255 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) G(n) = 2 2 -1/12 F(n) (F(n) - 2 F(n + 1)) (F(n) - 2 F(n) F(n + 1) - 1 + 2 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 256 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 20 2 14 177 217 2 G(n) = --- F(n + 1) - --- F(n + 1) - -- F(n) + --- F(n) F(n + 1) - --- F(n) 319 319 29 319 638 629 2 3 1209 3 2 113 4 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 638 638 319 525 5 + --- F(n) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 257 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 25 2 3 G(n) = - 5/22 + 5/22 F(n + 1) - -- F(n) F(n + 1) + 9/11 F(n) F(n + 1) 22 13 2 2 2 13 3 3 + -- F(n) F(n + 1) - F(n) F(n + 1) - -- F(n) - 1/11 F(n) F(n + 1) 11 22 4 + 9/11 F(n) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 258 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 2 2 G(n) = -1/12 F(n) (-3 + F(n + 1) - F(n) F(n + 1) - F(n) - 19 F(n) F(n + 1) 3 4 + 19 F(n) F(n + 1) + 4 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 259 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 2 20 3 245 5 648 5 G(n) = --- F(n + 1) - --- F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 319 461 2 1321 2 4 1699 3 3 - --- F(n) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 319 319 189 4 2 461 3 268 2 + --- F(n) F(n + 1) + --- F(n) + --- F(n) F(n + 1) 638 638 319 775 2 - --- F(n) F(n + 1) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 260 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 25 3 2 2 G(n) = 5/22 - 5/22 F(n + 1) + -- F(n) - F(n) F(n + 1) + 5/2 F(n) F(n + 1) 22 2 3 3 30 3 2 4 + 1/11 F(n) F(n + 1) - 3 F(n) F(n + 1) + -- F(n) F(n + 1) + F(n) 11 12 4 26 5 - -- F(n) F(n + 1) - -- F(n) 11 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 261 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 G(n) = 1/12 F(n + 1) (-71 F(n + 1) + 71 F(n + 1) + 14 F(n) 4 2 3 3 2 - 154 F(n) F(n + 1) - 48 F(n) F(n + 1) + 188 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 262 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 91 2 3 16 5 29 4 G(n) = -- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 22 63 2 4 79 3 2 10 4 2 - -- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 11 11 11 59 4 103 3 3 5 20 2 + -- F(n) F(n + 1) + --- F(n) F(n + 1) - 7 F(n) F(n + 1) + -- F(n) 11 11 11 2 20 5 - 5/22 F(n + 1) + 5/22 F(n + 1) - -- F(n) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 263 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 185 185 2 61 123 2 G(n) = --- F(n + 1) - --- F(n + 1) + 1/58 F(n) + --- F(n) F(n + 1) - --- F(n) 638 638 319 319 397 2 3 687 3 2 84 4 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 638 638 319 235 5 + --- F(n) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 264 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 571 2 16290 3 1674 3 1674 2 G(n) = ---- F(n) F(n + 1) + ----- F(n + 1) + ---- F(n) - ---- F(n) 551 6061 6061 6061 11070 2 5 2733 71515 6 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 6061 12122 42975 3 4 21 2 31965 7 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) - ----- F(n + 1) 6061 418 12122 615 2 - ----- F(n + 1) 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 265 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = -1/12 F(n) (-7 + 5 F(n + 1) - 5 F(n) F(n + 1) + 5 F(n) 2 2 3 4 - 17 F(n) F(n + 1) + 17 F(n) F(n + 1) + 2 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 266 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 185 3 129 3 571 1729 2 4 G(n) = ---- F(n + 1) + --- F(n) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 319 638 638 129 2 889 5 127 3 3 - --- F(n) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 319 257 2 123 2 181 2 547 6 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n + 1) + --- F(n + 1) 319 319 319 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 267 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 5 G(n) = 1/12 F(n) (6 F(n) - 6 F(n) - 8 F(n + 1) + 10 F(n + 1) 2 4 2 3 + 12 F(n) F(n + 1) - 19 F(n) F(n + 1) + 16 F(n) F(n + 1) 3 2 4 2 - 23 F(n) F(n + 1) + 28 F(n) F(n + 1) - 16 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 268 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 185 3 252 3 4 3 G(n) = --- F(n + 1) + --- F(n) - 1/2 F(n) + 2 F(n) F(n + 1) 638 319 4095 3 4 479 2 765 6 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 319 2615 2 5 125 4 3 2 2 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) - 2 F(n) F(n + 1) 319 638 1293 2 185 + ---- F(n) F(n + 1) - --- 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 269 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 145 7 11 145 3 11 3 G(n) = ---- F(n + 1) - -- F(n) + --- F(n + 1) + -- F(n) 24 12 24 12 3 4 2 2 - 5/2 F(n) F(n + 1) + 79/6 F(n) F(n + 1) - 4 F(n) F(n + 1) 3 2 4 955 4 3 - 25/6 F(n) F(n + 1) + 25/6 F(n) F(n + 1) - --- F(n) F(n + 1) 24 2 5 + 265/8 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 270 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 31965 4 3 11010 3 4 991 2 G(n) = ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 12122 6061 12122 39 2 7585 6 9825 2 5 - --- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 209 12122 12122 2226 2183 3 2183 2 1070 2 - ---- F(n) F(n + 1) - ---- F(n) + ---- F(n) + ---- F(n + 1) 6061 6061 6061 6061 1070 3 - ---- F(n + 1) 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 271 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 5 3 4 G(n) = -1/924 F(n) (13 F(n) - 2350 F(n) F(n + 1) + 11650 F(n) F(n + 1) 6 3 7 - 10425 F(n) F(n + 1) - 4698 F(n + 1) + 4550 F(n + 1) 2 2 3 + 1547 F(n) F(n + 1) - 248 F(n) F(n + 1) - 26 F(n + 1) - 13 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 272 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 25 2 427 2 20 3 13 3 G(n) = -- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n + 1) + --- F(n) 58 638 319 319 683 6 207 5 929 3 3 - --- F(n) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 58 638 1729 4 2 229 5 20 2 670 2 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n + 1) + --- F(n) 638 638 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 273 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2798 2 4457 2 2 24439 3 G(n) = ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 6061 6061 1631 10575 7 399725 3 5 + ----- + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 12122 319 12122 23050 2 6 3379 3 167279 4 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + ------ F(n + 1) 6061 6061 12122 7775 8 2734 2 1070 3 1631 3 - ---- F(n + 1) - ---- F(n) F(n + 1) + ---- F(n + 1) - ----- F(n) 551 6061 6061 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 274 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 243 3 4 20 20 3 537 6 G(n) = --- F(n) - 7/22 F(n) - --- + --- F(n + 1) - --- F(n) F(n + 1) 638 319 319 319 75 2 3 2 2 - --- F(n) F(n + 1) + 2/11 F(n) F(n + 1) - F(n) F(n + 1) 319 12 3 548 2 5 3477 3 4 + -- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 319 638 4009 4 3 309 5 2 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 275 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 1269 3 5 9543 4 4 7 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + 9/638 F(n) F(n + 1) 58 638 20 37 20 1757 2 2 + --- + --- F(n) - --- F(n + 1) - ---- F(n) F(n + 1) 319 638 319 319 12 2 3 53 4 437 2 6 - -- F(n) F(n + 1) - -- F(n) F(n + 1) + --- F(n) F(n + 1) 11 22 58 61 3 2 1909 3 309 4 + -- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) 22 638 638 1709 5 3 5 + ---- F(n) F(n + 1) + 4/11 F(n) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 276 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 257517 2 22007675 3 6 2925909 4 G(n) = -------- F(n + 1) + -------- F(n) F(n + 1) + ------- F(n) F(n + 1) 1206139 2412278 4824556 503145 5 515034 3 3 - ------- F(n) F(n + 1) + ------- F(n) F(n + 1) 1206139 1206139 30579925 5 4 2821813 3 2 - -------- F(n) F(n + 1) - ------- F(n) F(n + 1) 2412278 2412278 19528511 2 3 5092750 2 7 11889 6 + -------- F(n) F(n + 1) - ------- F(n) F(n + 1) + ------- F(n + 1) 4824556 1206139 2412278 257517 6 257517 5 462249 5 544041 - ------- F(n) + ------- F(n) + ------- F(n + 1) + ------- F(n + 1) 1206139 1206139 4824556 4824556 10328575 4 5 574479 4 2 + -------- F(n) F(n + 1) + ------- F(n) F(n + 1) 2412278 2412278 574479 2 4 - ------- F(n) F(n + 1) 2412278 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 277 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 615 3 540 7 58555 5 3 G(n) = ------ F(n + 1) - ---- F(n) - ----- F(n) F(n + 1) 12122 6061 12122 2214 2 2712 2 1080 6 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 6061 6061 2155 2 2 8972 7 540 6 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 12122 6061 6061 41691 2 6 1080 2 5 3367 7 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 12122 6061 12122 7831 6 2 16315 3 465 8 615 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) + ----- 6061 12122 12122 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 278 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 6 6 G(n) = 1/1848 F(n) (-423 F(n + 1) + 141 F(n + 1) - 488 F(n) 4 2 3 3 4 4 + 1193 F(n) F(n + 1) + 976 F(n) F(n + 1) + 6818 F(n) F(n + 1) 5 5 3 6 2 - 694 F(n) F(n + 1) - 15036 F(n) F(n + 1) + 1682 F(n) F(n + 1) 7 2 2 2 4 + 894 F(n) F(n + 1) + 2478 F(n) F(n + 1) - 1193 F(n) F(n + 1) 4 8 2 6 8 + 1260 F(n + 1) - 630 F(n + 1) + 2534 F(n) F(n + 1) + 488 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 279 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(4 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 60833 4 2 2 G(n) = 3/11 F(n) F(n + 1) + ----- F(n) F(n + 1) - 9/22 F(n) F(n + 1) 60610 2 6 571569 6 3 3 - 3/22 F(n) F(n + 1) + ------ F(n) F(n + 1) + 6/11 F(n) F(n + 1) 60610 4677 2 7 8 35344 8 1134 9 + ---- F(n) F(n + 1) - 3/22 F(n) - ----- F(n) F(n + 1) + ---- F(n) 638 30305 6061 10891 7 2 2884 8 130591 3 6 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 6061 6061 12122 7 276079 2 3 615 9 615 - 3/11 F(n) F(n + 1) - ------ F(n) F(n + 1) + ----- F(n + 1) - ----- 60610 12122 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 280 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 25 3 2 2 G(n) = 5/22 - 5/22 F(n + 1) + -- F(n) - F(n) F(n + 1) + 2 F(n) F(n + 1) 22 13 2 3 3 27 3 2 + -- F(n) F(n + 1) - 2 F(n) F(n + 1) + -- F(n) F(n + 1) 22 22 13 4 15 5 - -- F(n) F(n + 1) - -- F(n) 22 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 281 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(n + j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 29 2 4 6 29 6 G(n) = 1/3 + -- F(n + 1) - 1/3 F(n + 1) - 1/3 F(n) - -- F(n + 1) 60 60 41 3 5 - -- F(n) F(n + 1) + 1/3 F(n) F(n + 1) + 17/6 F(n) F(n + 1) 30 2 2 2 4 3 + 5/6 F(n) F(n + 1) - 7/2 F(n) F(n + 1) - 3/2 F(n) F(n + 1) 27 5 + -- F(n) F(n + 1) 10 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 282 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(n + j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 4 125 7 890 6 G(n) = ---- F(n + 1) + --- F(n + 1) - --- F(n) F(n + 1) 319 638 319 3845 3 4 107 3 85 3 107 4 - ---- F(n) F(n + 1) + --- F(n) - --- F(n + 1) - --- F(n) 638 319 638 319 1119 2 457 3 581 2 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 638 638 638 560 2 2 5105 2 5 645 3 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 319 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 283 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(n + j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 85 3 3 26 4 53 4 2 G(n) = -- F(n) F(n + 1) + -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 22 5 29 4 16 5 - 7/2 F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 69 2 3 41 2 4 103 3 2 + -- F(n) F(n + 1) - -- F(n) F(n + 1) - --- F(n) F(n + 1) 22 11 22 2 2 5 - 5/22 F(n + 1) + 5/22 F(n + 1) + 9/11 F(n) - 9/11 F(n) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 284 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 3 3 G(n) = -1/12 F(n) (-9 F(n) + 9 F(n) - 6 F(n) F(n + 1) - 18 F(n) F(n + 1) 4 2 2 4 3 3 + 3 F(n + 1) - 150 F(n + 1) - 115 F(n) F(n + 1) + 405 F(n) F(n + 1) 2 2 6 + 21 F(n) F(n + 1) + 145 F(n + 1) + 40 F(n) F(n + 1) 5 - 325 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 285 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 435 4 3 84 G(n) = 5/22 F(n + 1) + --- F(n) F(n + 1) - -- F(n) F(n + 1) 11 11 445 6 75 2 1065 2 5 + --- F(n) F(n + 1) + -- F(n) F(n + 1) - ---- F(n) F(n + 1) 22 11 22 3 3 75 3 4 23 2 3 - 31/2 F(n) F(n + 1) + -- F(n) F(n + 1) + -- F(n) - 5/22 F(n + 1) 22 22 23 3 203 2 5 2 4 - -- F(n) - --- F(n) F(n + 1) + 6 F(n) F(n + 1) - 23/2 F(n) F(n + 1) 22 11 4 2 + 51/2 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 286 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1031 2 193 7 1277 6 G(n) = ---- F(n) F(n + 1) + --- F(n + 1) - ---- F(n) F(n + 1) 319 319 638 201 3 185 3 3323 5 2 3071 6 - --- F(n + 1) + --- F(n) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 638 638 638 2 2 5105 2 5 3 185 - 3/2 F(n) F(n + 1) + ---- F(n) F(n + 1) + 2 F(n) F(n + 1) - --- 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 287 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(n + 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 3 5 G(n) = 1/12 F(n) (3 F(n) + 14 F(n) F(n + 1) - 86 F(n) F(n + 1) 2 2 3 4 2 - 27 F(n) F(n + 1) + 20 F(n) F(n + 1) - 3 F(n) - 34 F(n + 1) 4 2 5 4 - 9 F(n) F(n + 1) + 115 F(n) F(n + 1) - 29 F(n) F(n + 1) - 7 F(n + 1) 6 + 43 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 288 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(n + 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 9525 4 4 443 5 4825 2 6 G(n) = ---- F(n) F(n + 1) + --- F(n) + ---- F(n) F(n + 1) 638 638 638 6975 3 5 850 5 3 185 314 4 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) + --- - --- F(n) 319 319 638 319 185 3 62 4 - --- F(n + 1) + 9/638 F(n) F(n + 1) + --- F(n) F(n + 1) 638 319 3195 2 2 762 2 3 795 3 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 638 319 319 1214 3 2 1347 4 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 289 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 3 3 6 G(n) = 47/6 F(n) F(n + 1) - 11/2 F(n) F(n + 1) + 1/12 F(n + 1) 2 8 4 6 - 1/12 F(n + 1) - 1 - 1/2 F(n + 1) + 5/12 F(n) + 7/12 F(n) 4 5 3 4 4 + 3/2 F(n + 1) + 7 F(n) F(n + 1) - 71/4 F(n) F(n + 1) 5 4 2 3 5 - 5/3 F(n) F(n + 1) + 7 F(n) F(n + 1) + 77/6 F(n) F(n + 1) 6 2 - 39/4 F(n) F(n + 1) - F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 290 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 268886 2 3 2 6 115490 2 7 G(n) = ------ F(n) F(n + 1) - 1/2 F(n) F(n + 1) - ------ F(n) F(n + 1) 30305 18183 552616 4 179794 7 2 513325 8 - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 90915 18183 36366 6 2 2207072 6 3 1070 9 - 2/11 F(n) F(n + 1) + ------- F(n) F(n + 1) - ---- F(n + 1) 90915 6061 15 8 3 16 7 7 - -- F(n) - 1/11 F(n) F(n + 1) - -- F(n) F(n + 1) - 1/11 F(n) F(n + 1) 22 11 5158 8 13 2 2 1070 123787 9 + ---- F(n) F(n + 1) + -- F(n) F(n + 1) + ---- - ------ F(n) 1595 11 6061 36366 71081 15 3 5 + ----- F(n) + -- F(n) F(n + 1) 18183 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 291 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(n + 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 13 4 2847 2 2 2 3 G(n) = --- F(n) F(n + 1) - ---- F(n) F(n + 1) - 2/11 F(n) F(n + 1) 22 638 882 3 85 3 2 93 4 + --- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 319 22 22 850 7 8225 2 6 6125 4 4 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 638 638 675 20 20 65 4 724 3 + --- F(n) - --- F(n + 1) + --- - -- F(n) + --- F(n) F(n + 1) 638 319 319 58 319 6125 3 5 - ---- F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 292 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(n + 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 25602 5 889 5339 126925 3 6 G(n) = ----- F(n + 1) + --- F(n) - ---- F(n + 1) - ------ F(n) F(n + 1) 319 638 638 638 4544 2 3 111 2 4 20045 4 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 11 638 99875 8 2473 7763 2 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n + 1) 638 319 319 267 6 9233 3 2 18575 2 7 + --- F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 11 319 319 1231 5 651 3 3 889 2 45825 9 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) - ----- F(n + 1) 22 11 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 293 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2185 6 161 2 2 4 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) + 12 F(n) F(n + 1) 22 22 195 2 5 3 3 5 45 3 - --- F(n) F(n + 1) - 72 F(n) F(n + 1) + 66 F(n) F(n + 1) - -- F(n) 22 22 435 7 6 203 2 865 3 - --- F(n + 1) - 29 F(n + 1) - --- F(n) F(n + 1) + --- F(n + 1) 11 11 22 1665 3 4 643 2 45 2 106 - ---- F(n) F(n + 1) + --- F(n + 1) + -- F(n) - --- F(n) F(n + 1) 22 22 22 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 294 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(2 n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 1891 4 8 2 G(n) = -29/2 F(n + 1) - ---- F(n + 1) + 475/3 F(n + 1) + 29/2 F(n + 1) 12 2 25 2 6 3 + 3/4 F(n) - -- F(n) F(n + 1) - 9/2 F(n) F(n + 1) + 15/2 F(n) F(n + 1) 12 3 3 7 2 2 - 36 F(n) F(n + 1) - 3/4 - 1150/3 F(n) F(n + 1) - 74/3 F(n) F(n + 1) 2 4 3 5 5 + 6 F(n) F(n + 1) + 675/2 F(n) F(n + 1) + 33 F(n) F(n + 1) 3 + 395/6 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 295 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(2 n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 6 3 63 4 G(n) = -5/22 F(n + 1) + 55 F(n) F(n + 1) - 82 F(n) F(n + 1) + -- F(n) 22 3 925 7 377 2 + 5/22 F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 11 22 835 2 2 4 3 4 4 + --- F(n) F(n + 1) + 145 F(n) F(n + 1) + 325/2 F(n) F(n + 1) 22 2 5 4575 2 6 3 4 - 125 F(n) F(n + 1) - ---- F(n) F(n + 1) - 35 F(n) F(n + 1) 22 1253 2 155 3 425 3 5 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 22 11 22 63 3 - -- F(n) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 296 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 185 2 947 2 185 9 G(n) = -47/2 F(n) F(n + 1) - --- F(n + 1) - --- F(n) + --- F(n + 1) 638 638 638 7855 8 333 4 34905 3 6 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 22 638 2597 4 23005 4 5 4430 2 7 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 4039 3 2 3 3 177 2 3 + ---- F(n) F(n + 1) + 11 F(n) F(n + 1) - --- F(n) F(n + 1) 319 29 2357 2 4 5 947 + ---- F(n) F(n + 1) + 13 F(n) F(n + 1) - 7 F(n) F(n + 1) + --- F(n) 319 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 297 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(2 n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 6 G(n) = -1/12 F(n) (339 F(n + 1) + 18 F(n) - 330 F(n + 1) 5 2 2 2 6 + 750 F(n) F(n + 1) - 550 F(n) F(n + 1) - 300 F(n) F(n + 1) 3 3 4 8 - 870 F(n) F(n + 1) - 3868 F(n + 1) + 3875 F(n + 1) - 99 F(n) F(n + 1) 3 7 2 4 + 1561 F(n) F(n + 1) - 9325 F(n) F(n + 1) + 210 F(n) F(n + 1) 3 3 5 + 157 F(n) F(n + 1) + 8450 F(n) F(n + 1) - 18) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 298 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 21470 7 1384 2 2 23230 7 G(n) = ------ F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 231 231 231 5493 7 2 835 8 7067 3 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 220 22 77 2273 4 603 8 2987 2 3 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 44 44 110 8240 2 6 2583 2 7 7507 3 2 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 77 110 220 4558 222 851 4 43 9 439 5 + ---- - --- F(n + 1) - --- F(n + 1) - -- F(n) + --- F(n + 1) 231 11 22 11 22 3655 8 4430 8 - ---- F(n) + ---- F(n + 1) 231 231 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 299 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(2 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 5 G(n) = 1/6 F(n) (-5 F(n) F(n + 1) - 2 F(n + 1) + 6 F(n + 1) 2 4 2 3 + 3 F(n) F(n + 1) - 10 F(n) F(n + 1) + 6 F(n) F(n + 1) 3 2 4 3 - 14 F(n) F(n + 1) + 22 F(n) F(n + 1) - 5 F(n) - F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 300 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(3 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 415 4 3 80 6 175 2 5 G(n) = --- F(n) F(n + 1) + -- F(n) F(n + 1) - --- F(n) F(n + 1) 22 11 11 95 3 4 3 69 2 - -- F(n) F(n + 1) + 1/11 F(n) F(n + 1) + -- F(n) F(n + 1) - 5/22 22 22 169 2 21 2 2 3 - --- F(n) F(n + 1) - -- F(n) F(n + 1) + 3/11 F(n) F(n + 1) 22 22 3 25 3 4 + 5/22 F(n + 1) - -- F(n) + 4/11 F(n) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 301 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(3 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 3 3 3 G(n) = -1/6 F(n) (110 F(n + 1) + 290 F(n) F(n + 1) - 6 F(n) F(n + 1) 5 2 2 3 - 250 F(n) F(n + 1) - 7 F(n) - 117 F(n + 1) - 2 F(n) F(n + 1) 2 2 2 4 4 + 11 F(n) F(n + 1) - 70 F(n) F(n + 1) + 3 F(n + 1) + 37 F(n) F(n + 1) 4 + F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 302 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 105 3 345 3 120 4 189 2 G(n) = ---- F(n + 1) - --- F(n) + --- F(n) + --- F(n) F(n + 1) 638 638 319 638 1130 3 5 105 4920 3 2 - ---- F(n) F(n + 1) + --- - ---- F(n) F(n + 1) - 3/29 F(n) F(n + 1) 319 638 319 2012 2 2 24 3 4910 7 + ---- F(n) F(n + 1) - -- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 11 319 11515 2 6 1030 4 4 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 29 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 303 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 64 5 37 4 168 3 5 G(n) = -9/10 F(n + 1) - -- F(n) + -- F(n) F(n + 1) - --- F(n) F(n + 1) 55 22 11 94 5 3 45 4 181 4 4 + -- F(n) F(n + 1) - -- F(n) F(n + 1) + --- F(n) F(n + 1) 11 22 11 6 2 98 7 34 37 - 1/11 F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) + -- F(n + 1) 11 55 55 4 23 4 15 21 8 + 2/11 F(n) - -- F(n + 1) + -- + -- F(n + 1) 11 11 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 304 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 5 7 G(n) = 17/3 F(n + 1) + 470 F(n) F(n + 1) - 1630/3 F(n) F(n + 1) 2 2 2 4 - 39 F(n) F(n + 1) + 4 F(n) F(n + 1) - 5/3 F(n) F(n + 1) - 2 5 8 2 6 + 37/3 F(n) F(n + 1) + 670/3 F(n + 1) + 5 F(n) F(n + 1) 6 3 3 - 17/3 F(n + 1) + 94 F(n) F(n + 1) + 46/3 F(n) F(n + 1) 3 3 4 - 50/3 F(n) F(n + 1) - 664/3 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 305 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 15 3 G(n) = - 5/22 + 5/22 F(n + 1) + -- F(n) - 9/11 F(n) F(n + 1) 22 21 2 2 2 3 15 3 + -- F(n) F(n + 1) + F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 3 2 4 4 + F(n) F(n + 1) + 6/11 F(n) - 2 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 306 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 3 3 2 G(n) = 1/12 F(n) (3 F(n) - 14 F(n) F(n + 1) - 80 F(n) F(n + 1) 4 2 3 5 + 112 F(n) F(n + 1) - 8 F(n) F(n + 1) - 4 F(n + 1) + 10 F(n + 1) 2 4 + 6 F(n) F(n + 1) + 2 F(n) F(n + 1) - 27 F(n)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 307 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 81 4 10919 4 3 295 7 393 G(n) = --- F(n + 1) - ----- F(n) F(n + 1) - ---- F(n + 1) + --- 58 1276 1276 319 505 3 1696 3 249 3 4 1503 6 + ---- F(n + 1) - ---- F(n) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 1276 319 319 638 4493 2 5 954 3 414 3 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 1276 319 319 1161 4 5681 7 - ---- F(n) + ---- F(n) 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 308 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 42 3 2 19 3 3 34 4 G(n) = --- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 11 22 11 25 4 2 85 5 4 - -- F(n) F(n + 1) + -- F(n) F(n + 1) - F(n) F(n + 1) 11 22 19 5 41 2 3 2 4 23 5 + -- F(n) F(n + 1) + -- F(n) F(n + 1) + 3/11 F(n) F(n + 1) - -- F(n) 22 22 22 21 2 2 - -- F(n) - 5/22 F(n + 1) + 5/22 F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 309 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 6 2 G(n) = -1/12 F(n) (-92 F(n) + 8 F(n) + 48 F(n) - 13 F(n + 1) 5 2 2 3 + 13 F(n) F(n + 1) + 22 F(n) F(n + 1) - 14 F(n) F(n + 1) 4 2 5 3 3 - 226 F(n) F(n + 1) + 303 F(n) F(n + 1) - 41 F(n) F(n + 1) 3 4 - 15 F(n) F(n + 1) + 7 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 310 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 107 6 287 4 2 5 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) - 14 F(n) F(n + 1) 44 22 5 2 13 5 23 6 + 7 F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 11 22 51 2 23 2 4 2 83 2 - -- F(n) F(n + 1) - -- F(n) F(n + 1) - 5/22 F(n + 1) + -- F(n) 44 22 22 3 2 5 197 4 3 18 7 + 5/22 F(n + 1) + 9/44 F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) 44 11 53 6 - -- F(n) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 311 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(3 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 2 6 G(n) = -1/84 F(n + 1) F(n) (-2055 F(n + 1) F(n) - 82 F(n) + 870 F(n + 1) 3 3 4 2 + 381 F(n + 1) F(n) + 2010 F(n + 1) F(n) - 210 F(n + 1) F(n) 2 - 914 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 312 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - 3 j) F(3 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 217 5 2 1085 6 1732 5 3 291 G(n) = ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- 638 1276 319 1276 446 6 2 177 7 79 7 + --- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 638 638 955 2 367 4 4 217 2 5 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 1276 116 319 93 3 27 3 1097 8 107 8 159 4 - --- F(n) + --- F(n + 1) + ---- F(n) - ---- F(n + 1) + --- F(n + 1) 319 116 1276 1276 638 217 7 217 7 - ---- F(n + 1) - --- F(n) 1276 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 313 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - 3 j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 690 747455 4 5 420340 2 7 G(n) = ---- F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 6061 12122 6061 8515 3 6 1818 2 690 2 - ----- F(n) F(n + 1) - ---- F(n) - ---- F(n + 1) 12122 6061 6061 177054 4 160411 2 3 59965 3 2 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 12122 12122 4591 4 172420 8 690 9 1818 + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) + ---- F(n + 1) + ---- F(n) 6061 6061 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 314 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - 3 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 5 3 G(n) = -1/84 F(n) (-2 F(n) F(n + 1) + 1725 F(n) F(n + 1) 2 6 3 4 4 + 525 F(n) F(n + 1) + 242 F(n) F(n + 1) - 675 F(n) F(n + 1) 2 3 5 2 4 + 3 F(n + 1) - 1125 F(n) F(n + 1) + 5 F(n) F(n + 1) 2 2 6 4 2 - 692 F(n) F(n + 1) - F(n + 1) + 27 F(n) + 19 F(n) - 46 4 2 - 5 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 315 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - 3 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 555 4 2213 8 2213 8 360751 5 G(n) = ---- F(n + 1) + ---- F(n) + ---- F(n + 1) - ------ F(n + 1) 638 957 957 130790 14718 9 2641 201821 2257 - ----- F(n) - ---- + ------ F(n) - ------ F(n + 1) 5945 1914 130790 130790 139295 8 4426 6 2 1211341 6 3 - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) + ------- F(n) F(n + 1) 26158 1595 65395 11065 7 854429 7 2 27656 8 + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 1914 130790 5945 64638 4 177404 9 11511 3 - ----- F(n) F(n + 1) + ------ F(n + 1) - ----- F(n) F(n + 1) 65395 65395 3190 42047 7 14158 2 2 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 9570 4785 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 316 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(3 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = -1/12 F(n) (-7 + F(n + 1) - F(n) F(n + 1) + 7 F(n) 2 2 3 - 25 F(n) F(n + 1) + 25 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 317 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - 3 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 20 134 157 3 G(n) = - --- + --- F(n + 1) + --- F(n) - --- F(n) F(n + 1) 319 319 319 319 139 2 2 861 2 3 439 3 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 638 638 638 252 3 2 59 4 217 4 287 5 - --- F(n) F(n + 1) + --- F(n) + --- F(n) F(n + 1) - --- F(n) 319 638 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 318 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - 3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 57987 58536 2 5397 5 57846 6 G(n) = ----- F(n + 1) + ----- F(n + 1) - ---- F(n + 1) - ----- F(n + 1) 12122 6061 1102 6061 11805 11805 2 2419 567 4 - ----- F(n) + ----- F(n) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 12122 12122 638 58 288113 5 2688 2 3 1791 2 4 + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 12122 551 12122 16191 3 2 243481 3 3 - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 1102 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 319 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - 3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 2 4 29 3 G(n) = 28/3 F(n + 1) - 17/4 F(n) F(n + 1) - -- F(n) F(n + 1) 12 37 2 2 2 2 + -- F(n) F(n + 1) + 5/6 F(n) F(n + 1) - 28/3 F(n + 1) - 7/12 F(n) 12 4 281 3 3 253 5 - 7/12 F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 12 12 3 + F(n) F(n + 1) + 7/12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 320 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - 3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 20 2 81 6 115 5 1351 2 G(n) = ---- F(n + 1) + --- F(n + 1) - --- F(n) - --- F(n) + ---- F(n) 319 319 116 319 1276 753 2 4 49 3 2 331 4 - ---- F(n) F(n + 1) - -- F(n) F(n + 1) + --- F(n) F(n + 1) 1276 29 638 577 4 2 1237 5 41 4 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) - -- F(n) F(n + 1) 116 319 58 405 5 235 2 3 + --- F(n) F(n + 1) + --- F(n) F(n + 1) 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 321 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 6 4 3 G(n) = 1/24 F(n) (-28 F(n) - 9 F(n + 1) - 6 F(n + 1) + 12 F(n) F(n + 1) 2 2 2 4 3 - 52 F(n) F(n + 1) + 153 F(n) F(n + 1) + 46 F(n) F(n + 1) 3 3 5 4 2 - 216 F(n) F(n + 1) - 82 F(n) F(n + 1) + 127 F(n) F(n + 1) 2 6 + 27 F(n + 1) + 28 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 322 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4217 6 62 2 243 G(n) = ----- F(n) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 3190 145 1595 3533 5 5 2 2319 4 2 - ---- F(n) F(n + 1) - 10 F(n) F(n + 1) + ---- F(n) F(n + 1) 638 638 5283 4 3 673 6 164 2 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 638 1595 145 8177 6 20 2 2731 2 1713 5 + ---- F(n) F(n + 1) - --- F(n + 1) + ---- F(n) - ---- F(n) F(n + 1) 3190 319 1595 1595 20 3 249 7 + --- F(n + 1) - --- F(n) 319 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 323 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 317 5 47 6 6 47 2 G(n) = ---- F(n) F(n + 1) + -- F(n + 1) + 7/4 F(n) - -- F(n + 1) 60 30 30 4 2 245 4 4 5 3 + 21/4 F(n) F(n + 1) - --- F(n) F(n + 1) + 45/4 F(n) F(n + 1) 12 6 2 125 7 11 - 35/3 F(n) F(n + 1) + --- F(n) F(n + 1) + -- F(n) F(n + 1) 12 30 4 11 8 8 157 3 5 + 5/4 F(n + 1) - -- F(n) - 5/12 F(n + 1) - 5/6 + --- F(n) F(n + 1) 12 12 5 - 23/6 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 324 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - 2 j) F(3 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 25 4 41 3 27 5 37 8 G(n) = --- F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) - -- F(n) + 5/22 44 44 44 88 23 4 835 4 4 373 6 2 - -- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 22 88 88 285 7 25 109 4 - --- F(n) F(n + 1) - 5/22 F(n + 1) + -- F(n) + --- F(n) 44 44 88 3 2 47 7 343 3 5 + 5/4 F(n) F(n + 1) + -- F(n) F(n + 1) - --- F(n) F(n + 1) 44 44 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 325 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - 2 j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 47 3 2 5 7275 8 G(n) = --- F(n) - 4/21 F(n) F(n + 1) - 716/7 F(n + 1) - ---- F(n) F(n + 1) 84 28 375 2 7 1181 3 2 3 6 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) + 425/2 F(n) F(n + 1) 28 84 1475 9 1783 4 2 + ---- F(n + 1) + ---- F(n) F(n + 1) + 4/21 F(n) F(n + 1) 14 42 271 2 3 47 43 - --- F(n) F(n + 1) + -- F(n) - -- F(n + 1) 12 84 14 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 326 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - 2 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 412 197 8 555 8 41 8 G(n) = ---- F(n + 1) - --- F(n) + ---- F(n + 1) - --- F(n) F(n + 1) 957 348 1276 957 355 4 401 4 1733 4 5 - --- F(n + 1) - 5/3828 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 957 638 319 553 5 4 422 6 2 3317 7 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 66 957 1914 2741 7 2 3011 8 5747 5 32 5 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) + -- F(n + 1) 638 319 1914 87 4919 9 1895 7 811 4 4 + ---- F(n) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 1914 1914 1276 1916 5 3 + ---- F(n) F(n + 1) 957 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 327 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - 2 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 9 18 2 4 25 4 2 G(n) = 5/22 F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 11 22 4215 4 5 24 3 3 115 3 6 + ---- F(n) F(n + 1) + -- F(n) F(n + 1) - --- F(n) F(n + 1) 11 11 22 2 2 102 4 5 + 2/11 F(n) - 5/22 F(n + 1) + --- F(n) F(n + 1) - 4/11 F(n) F(n + 1) 11 5 1673 2 3 4740 2 7 - 1/11 F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 22 11 315 3 2 24 5 1965 4 - --- F(n) F(n + 1) - -- F(n) - ---- F(n) F(n + 1) 11 11 11 3915 8 + ---- F(n) F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 328 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(3 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 G(n) = - 5/22 + 5/22 F(n + 1) + 4/11 F(n) - 1/2 F(n) F(n + 1) 21 2 3 19 3 2 4 12 4 + -- F(n) F(n + 1) - -- F(n) F(n + 1) - 1/2 F(n) + -- F(n) F(n + 1) 11 11 11 5 - 7/11 F(n) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 329 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - 2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1128 3392 4 463 5 G(n) = ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 22 1704 2 3 53 2 4 5064 3 2 + ---- F(n) F(n + 1) - -- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 22 319 240 3 3 1668 5847 2 1688 5 + --- F(n) F(n + 1) + ---- F(n + 1) - ---- F(n + 1) - ---- F(n + 1) 11 319 638 319 203 6 681 595 2 + --- F(n + 1) - --- F(n) - --- F(n) 22 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 330 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - 2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 13 75 3 7 76 3 G(n) = --- F(n) + -- F(n + 1) - 487/4 F(n + 1) + 487/4 F(n + 1) + -- F(n) 21 28 21 225 3 2 1457 2 75 2 3 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) + -- F(n) F(n + 1) 28 84 28 2 5 75 4 713 2 - 29/6 F(n) F(n + 1) + -- F(n) F(n + 1) + --- F(n) F(n + 1) 14 14 75 5 3143 3 4 6 - -- F(n + 1) + ---- F(n) F(n + 1) - 881/3 F(n) F(n + 1) 28 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 331 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - 2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 21 2 4 10 3 2 G(n) = 2/11 F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 49 3 3 15 4 87 4 2 - -- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 22 22 47 5 13 4 5 10 5 - -- F(n) F(n + 1) - -- F(n) F(n + 1) + 5/11 F(n) F(n + 1) + -- F(n) 22 22 11 2 2 + 1/11 F(n) - 5/22 F(n + 1) + 5/22 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 332 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 791 745 2 219 2 5 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 638 638 110 2914 3 3 179 3 4 20 2 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n + 1) 319 22 319 1593 5 104 6 180 2 4 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 11 319 3788 2 7 20 3 5947 3 + ---- F(n) F(n + 1) - 3/10 F(n) + --- F(n + 1) + ---- F(n) 1595 319 3190 3691 4 2 458 6 + ---- F(n) F(n + 1) + --- F(n) 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 333 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 30 3 3 5 2 51 2 G(n) = --- F(n) F(n + 1) - 35/2 F(n) F(n + 1) + -- F(n) F(n + 1) 11 11 2 5 15 2 - 5/2 F(n) F(n + 1) - -- F(n) F(n + 1) + 6/11 F(n) F(n + 1) 22 4 3 49 5 25 2 4 + 10 F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 22 22 18 4 2 3 4 18 6 2 + -- F(n) F(n + 1) + 15/2 F(n) F(n + 1) - -- F(n) - 5/22 F(n + 1) 11 11 3 3 + 5/22 F(n + 1) - 4/11 F(n) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 334 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 19 47 8 5 2 28 7 31 7 G(n) = -- + -- F(n + 1) + 4/55 F(n) F(n + 1) + -- F(n) - --- F(n + 1) 88 88 11 110 3 91 3 4 23 2 21 4 + 3/55 F(n + 1) - -- F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) 22 110 88 23 4 345 7 35 3 5 - -- F(n + 1) - --- F(n) F(n + 1) - -- F(n) F(n + 1) 44 44 11 32 4 3 145 4 4 5 3 + -- F(n) F(n + 1) + --- F(n) F(n + 1) - 63/4 F(n) F(n + 1) 11 11 53 6 1495 6 2 - -- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 88 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 335 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(3 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 2 6 3 G(n) = -1331/6 F(n + 1) + 1/2 F(n) + 5 F(n) F(n + 1) + 40/3 F(n) F(n + 1) 3 3 2 4 - 109/6 F(n) F(n + 1) + 7/2 F(n) F(n + 1) - 13/6 F(n) F(n + 1) 3 5 7 2 2 + 470 F(n) F(n + 1) - 1630/3 F(n) F(n + 1) - 77/2 F(n) F(n + 1) 5 3 6 + 89/6 F(n) F(n + 1) - 3/2 + 95 F(n) F(n + 1) - 20/3 F(n + 1) 8 2 + 670/3 F(n + 1) + 20/3 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 336 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3643 4 163298 8 4415 2 2 G(n) = ---- F(n) F(n + 1) + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 174 2871 2233 230906 2 3 2454 2 6 7 2 + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) - 827/9 F(n) F(n + 1) 2871 2233 6481 8 1549 2045 8 439 5 - ---- F(n) F(n + 1) - ---- + ---- F(n + 1) + --- F(n + 1) 99 4466 2233 29 87044 9 13004 9 24919 88111 - ----- F(n + 1) - ----- F(n) + ----- F(n) + ----- F(n + 1) 2871 957 1914 5742 234 4 10225 7 7771 7 2045 8 - --- F(n + 1) - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) 319 4466 4466 2233 39 3 + --- F(n) F(n + 1) 203 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 337 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 6 5 4 G(n) = -5/22 F(n + 1) + 2/11 F(n) + 1/11 F(n) + 1/11 F(n) F(n + 1) 191 3 6 9 14 130 4 - --- F(n) F(n + 1) + 5/22 F(n + 1) - -- F(n) + --- F(n) F(n + 1) 22 11 11 38 5 4 134 6 3 691 7 2 + -- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 11 11 22 83 4 2 166 4 5 39 3 3 - -- F(n) F(n + 1) + --- F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 11 5 19 5 27 - 9/11 F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 11 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 338 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 4 2 6 G(n) = 1/60 F(n) (1875 F(n) F(n + 1) + 82 F(n + 1) - 132 F(n + 1) 5 2 4 5 3 + 520 F(n) F(n + 1) - 730 F(n) F(n + 1) - 5232 F(n) F(n + 1) 6 2 7 5 + 2820 F(n) F(n + 1) + 1200 F(n) F(n + 1) + 346 F(n) F(n + 1) 4 8 3 6 + 546 F(n + 1) + 39 F(n + 1) - 1248 F(n) F(n + 1) - 40 F(n) 8 + 655 F(n) - 206 F(n) F(n + 1) - 495) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 339 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 144 3 2 3 3 G(n) = -7/2 F(n) F(n + 1) - --- F(n) F(n + 1) + 19 F(n) F(n + 1) 11 25 189 4 5 + -- F(n) F(n + 1) + --- F(n) F(n + 1) - 33/2 F(n) F(n + 1) 11 22 93 2 3 48 5 6 17 + -- F(n) F(n + 1) - -- F(n + 1) + 15/2 F(n + 1) - -- F(n) 22 11 22 91 80 2 2 + -- F(n + 1) - -- F(n + 1) - 5/22 F(n) 22 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 340 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 29 3 25 5 25 G(n) = -229/2 F(n + 1) + -- F(n) - -- F(n + 1) - 5/12 F(n) + -- F(n + 1) 12 12 12 559 2 7 25 2 3 + --- F(n) F(n + 1) + 229/2 F(n + 1) + -- F(n) F(n + 1) 12 12 2 5 3 2 4 - 28/3 F(n) F(n + 1) - 25/4 F(n) F(n + 1) + 25/6 F(n) F(n + 1) 2 3 4 6 - 59/4 F(n) F(n + 1) + 1501/6 F(n) F(n + 1) - 824/3 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 341 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 351 2 3 1359 4 14445 6 2 G(n) = ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 319 638 4639 7 3501 3 3992 2 6 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 638 319 1359 3 2 755 5 3 51 105 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) - --- F(n + 1) 319 319 58 638 105 1583 2 2 1835 7 351 4 + --- + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 638 638 319 638 1290 8 - ---- F(n) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 342 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 831 3 3 651 6 159 2 G(n) = ---- F(n + 1) + 7/22 F(n) - --- F(n + 1) + --- F(n) F(n + 1) 22 22 11 37 2 323 2 742 5 1603 3 3 + -- F(n) + --- F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 22 11 11 22 103 6 43 2 - --- F(n) F(n + 1) - 179/2 F(n) F(n + 1) - -- F(n) F(n + 1) 11 11 129 2 4 2 5 3 4 + --- F(n) F(n + 1) - 13/2 F(n) F(n + 1) + 173/2 F(n) F(n + 1) 11 7 + 38 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 343 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 6 407 5 11 2 2 G(n) = 185/6 F(n) F(n + 1) + --- F(n) F(n + 1) + -- F(n) + 89/6 F(n + 1) 12 12 3 4 59 + 1345/6 F(n) F(n + 1) - 2061/4 F(n + 1) - -- F(n) F(n + 1) 12 3 5 7 2 2 + 1075 F(n) F(n + 1) - 7615/6 F(n) F(n + 1) - 94 F(n) F(n + 1) 2 4 6 47 439 3 3 + 23/4 F(n) F(n + 1) - 89/6 F(n + 1) - -- - --- F(n) F(n + 1) 12 12 409 3 8 + --- F(n) F(n + 1) + 3115/6 F(n + 1) 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 344 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 5 5 3 5 2 17 4 G(n) = -15 F(n) F(n + 1) + 75 F(n) F(n + 1) - 55 F(n) F(n + 1) - -- F(n) 22 27 3 3 3 4 + 5/22 - -- F(n) - 5/22 F(n + 1) + 20 F(n) F(n + 1) 11 4 3 14 2 177 2 + 35 F(n) F(n + 1) - -- F(n) F(n + 1) + --- F(n) F(n + 1) 11 11 295 2 2 2 6 135 3 - --- F(n) F(n + 1) + 55/2 F(n) F(n + 1) + --- F(n) F(n + 1) 11 22 3 5 35 3 4 4 - 105/2 F(n) F(n + 1) + -- F(n) F(n + 1) - 55/2 F(n) F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 345 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 4 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 29728 2 159139 2 4 43718 9 G(n) = ------ F(n + 1) - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 18183 60610 90915 115011 5 954 6 1055119 6 4 - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) + ------- F(n) F(n + 1) 30305 6061 60610 477 6 954 2 5 477 7 113452 6 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) - ------ F(n) 6061 6061 6061 18183 1491 3 119356 10 18352 9 - ---- F(n) + ------ F(n) + ----- F(n) F(n + 1) 6061 18183 1653 196132 7 3 210887 5 5 31798 6 - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) + ----- F(n + 1) 18183 18183 18183 690 3 651 2 - ---- F(n + 1) - ---- F(n) F(n + 1) 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 346 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 4 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 355 7 47 10 67 8 283 4 G(n) = ---- F(n) F(n + 1) - -- F(n) - -- F(n) + --- F(n) 42 84 21 84 94 3 7 4 20 4 4 + -- F(n) F(n + 1) - 4/7 F(n + 1) + -- F(n) F(n + 1) 21 21 17 4 2 989 4 6 451 5 3 - -- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 30 140 42 611 5 677 9 1423 8 2 + --- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 105 210 84 391 7 3 167 3 5 25 2 10 6 + --- F(n) F(n + 1) - --- F(n) F(n + 1) - -- F(n + 1) + -- F(n + 1) 21 42 84 21 8 10 + 8/21 + 4/21 F(n + 1) - 5/28 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 347 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 4 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 133759 6 4 4149 6 3 3736 6 G(n) = ------ F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n + 1) 4785 1595 957 3753 7 2 1192 7 3 8 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - 9/29 F(n) F(n + 1) 638 957 13225 8 2 61 9 1422 4 + ----- F(n) F(n + 1) - -- F(n) F(n + 1) - ---- F(n) F(n + 1) 957 11 1595 14559 5 4907 9 12461 2 4 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 1595 4785 3190 24635 5 5 27 5 201 7157 2 - ----- F(n) F(n + 1) + -- F(n + 1) - --- F(n + 1) - ---- F(n + 1) 957 58 319 1914 354 6 354 9 21 8 7233 2 3 + --- F(n) - --- F(n) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 110 3190 1572 5 4 - ---- F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 348 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 74 5 105 3 345 3 619 5 G(n) = ---- F(n) F(n + 1) - --- F(n + 1) - --- F(n) - --- F(n) F(n + 1) 319 638 638 319 345 2 105 2 303 2 4 186 3 3 + --- F(n) + --- F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 638 638 319 319 1029 4 2 189 2 2 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) - 3/29 F(n) F(n + 1) 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 349 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 4 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 4 G(n) = 1/84 F(n) (-47 F(n) + 47 F(n) + 244 F(n) F(n + 1) 3 2 2 3 2 - 237 F(n) F(n + 1) + 106 F(n) F(n + 1) - 110 F(n) F(n + 1) 4 2 5 3 - 63 F(n) F(n + 1) + 48 F(n) F(n + 1) + 44 F(n + 1) - 32 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 350 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 4 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 15803 5 7599 3 2 10212 3 3 G(n) = ------ F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 6061 6061 96 4 53553 4 2 381 4 + --- F(n) F(n + 1) + ----- F(n) F(n + 1) - --- F(n) F(n + 1) 209 12122 551 3557 5 2424 2 3 4527 2 4 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 6061 12122 690 690 2 2118 5 2118 2 - ---- F(n + 1) + ---- F(n + 1) - ---- F(n) + ---- F(n) 6061 6061 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 351 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 4 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 731 2 105 7 33 4 105 4 G(n) = ---- F(n) F(n + 1) + --- F(n + 1) + -- F(n) - --- F(n + 1) 638 638 58 638 33 7 893 6 2951 2 5 72 6 - -- F(n) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) + -- F(n) F(n + 1) 58 638 638 29 41 2 273 3 339 2 2 + -- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 58 638 638 501 3 2795 3 4 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 352 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 6 6 G(n) = 1/84 F(n) (-94 F(n) + 94 F(n) + 359 F(n + 1) - 60 F(n) F(n + 1) 2 4 3 5 - 299 F(n + 1) - 48 F(n + 1) + 96 F(n) F(n + 1) - 718 F(n) F(n + 1) 2 2 3 4 2 - 221 F(n) F(n + 1) + 173 F(n) F(n + 1) + 965 F(n) F(n + 1) 5 - 247 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 353 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 4 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 100161 2 2268 3 15129 G(n) = ------- F(n) F(n + 1) - ---- F(n) - ----- F(n) F(n + 1) 6061 6061 6061 9535 6 26698 2 40735 2 5 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 551 6061 1102 26523 3 3 5590 3 4 49215 4 3 - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 551 1102 42813 4 2 9771 5 27504 2 4 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 6061 6061 2268 2 690 2 690 3 + ---- F(n) - ---- F(n + 1) + ---- F(n + 1) 6061 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 354 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 12193 13792 3 2010 3 5 15 2 6 G(n) = ----- - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - -- F(n) F(n + 1) 638 319 11 11 11485 4 43447 4 147 5 354 5 - ----- F(n) + ----- F(n + 1) + ---- F(n + 1) - --- F(n) 638 638 1276 319 357 1915 8 2330 7 - ---- F(n + 1) - ---- F(n + 1) + ---- F(n) F(n + 1) 1276 22 11 10955 2 2 1959 2 3 81 3 2 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 319 1276 319 1989 4 + ---- F(n) F(n + 1) 1276 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 355 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 4 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 5 2 G(n) = -1/420 F(n) (1648 F(n + 1) - 339 F(n) + 1770 F(n) F(n + 1) 7 4 3 4 + 705 F(n) - 366 F(n) - 11210 F(n) F(n + 1) + 4180 F(n) F(n + 1) 5 2 6 + 3995 F(n) F(n + 1) - 2990 F(n) F(n + 1) - 1328 F(n + 1) 7 4 3 4 - 600 F(n + 1) - 3090 F(n) F(n + 1) + 7405 F(n) F(n + 1) 3 + 220 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 356 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 4 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 535237 8 725847 9 1071 2 G(n) = ------- F(n) F(n + 1) - ------ F(n) + ---- F(n) 65395 130790 638 432713 8 1053447 4 2256 - ------ F(n) F(n + 1) - ------- F(n) F(n + 1) - ---- F(n) F(n + 1) 26158 130790 319 24866 7 2 6121469 6 3 414 2 4 - ----- F(n) F(n + 1) + ------- F(n) F(n + 1) - --- F(n) F(n + 1) 2255 130790 319 557853 9 492 6 26391 5 + ------ F(n + 1) - --- F(n + 1) + ----- F(n) F(n + 1) 65395 29 638 11481 3 3 253146 99327 10719 2 - ----- F(n) F(n + 1) + ------ F(n) - ----- F(n + 1) + ----- F(n + 1) 319 65395 65395 638 895527 5 - ------ F(n + 1) 130790 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 357 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 5 9 2 5 G(n) = -1/84 F(n) (-1364 F(n) F(n + 1) - 48 F(n + 1) - 46 F(n) F(n + 1) 5 4 5 2 6 3 + 1964 F(n) F(n + 1) + 8 F(n) F(n + 1) - 524 F(n) F(n + 1) 6 8 2 5 + 50 F(n) F(n + 1) + 148 F(n) F(n + 1) + 36 F(n) F(n + 1) - 202 F(n) 9 2 8 + 147 F(n) + 116 F(n) - 29 F(n) F(n + 1) - 250 F(n) F(n + 1) 6 7 7 4 + 15 F(n) F(n + 1) + 4 F(n + 1) + 23 F(n) - 48 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 358 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 3 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 200416 6 969 5 139 929 G(n) = ------ F(n + 1) + --- F(n + 1) + --- F(n) - --- F(n + 1) 319 638 319 638 9919 2 4 180 2 2274 + ---- F(n) F(n + 1) + --- F(n) - ---- F(n) F(n + 1) 58 319 319 2 3 528700 9 1194 4 + 2/29 F(n) F(n + 1) + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 77449 5 415400 3 7 1477 3 2 - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 638 45542 3 3 42475 2 8 11989 2 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n + 1) 319 319 319 7325 10 - ---- F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 359 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 4 G(n) = -1/12 F(n) (-15 F(n) + 2 F(n) + 25 F(n) + 40 F(n) F(n + 1) 2 2 3 3 2 + 10 F(n) F(n + 1) - 70 F(n) F(n + 1) + 20 F(n) F(n + 1) 3 5 2 + 4 F(n + 1) - 10 F(n + 1) - 6 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 360 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 3 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 19 5 54 6 21 359 27 2 G(n) = -- F(n) + --- F(n) - --- F(n) - --- F(n + 1) - --- F(n + 1) 22 319 638 638 110 5 983 6 353 3 3 + 1/2 F(n + 1) + ---- F(n + 1) - --- F(n) F(n + 1) 3190 319 4 2415 4 2 2502 5 - 5/11 F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 1595 42 37 4 + ---- F(n) F(n + 1) - -- F(n) F(n + 1) 1595 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 361 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4305 5 9390 3 8793 2 451 3 4 G(n) = ---- F(n) F(n + 1) + ---- F(n) + ---- F(n) + --- F(n) F(n + 1) 551 6061 6061 58 79395 2 1037 6 54540 - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 12122 58 6061 7560 3 3 2493 2 5 1308 4 3 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 551 58 29 229835 2 18375 2 4 15645 4 2 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 12122 1102 551 690 2 690 3 - ---- F(n + 1) + ---- F(n + 1) 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 362 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 2 3 G(n) = -1/12 F(n) (-145 F(n) F(n + 1) - 11 F(n + 1) - 10 F(n) F(n + 1) 4 2 2 4 3 2 - F(n) + 25 F(n) F(n + 1) + 5 F(n + 1) - 8 F(n) F(n + 1) - 78 F(n) 6 3 3 5 5 + 67 F(n) - 55 F(n) F(n + 1) + 200 F(n) F(n + 1) + 11 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 363 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1545 20 3 150 3 488 2 G(n) = ---- F(n) F(n + 1) + --- F(n + 1) - --- F(n) - --- F(n) 319 319 319 319 136 2 4 5459 2 24755 2 5 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 11 319 638 52 2 11465 6 3075 3 4 + -- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 638 319 3 3 14620 4 3 127 5 - 9/11 F(n) F(n + 1) + ----- F(n) F(n + 1) - --- F(n) F(n + 1) 319 22 255 4 2 20 2 - --- F(n) F(n + 1) - --- F(n + 1) 22 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 364 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 4 3 6 G(n) = -1/12 F(n) (-60 F(n) F(n + 1) - 322 F(n) F(n + 1) - 86 F(n) F(n + 1) 6 5 2 7 - 14 F(n) F(n + 1) + 105 F(n) F(n + 1) - 12 F(n + 1) 2 2 3 5 + 56 F(n) F(n + 1) + 39 F(n) + 126 F(n) F(n + 1) + 6 F(n + 1) 3 4 4 5 7 + 231 F(n) F(n + 1) - 54 F(n) F(n + 1) - 36 F(n) + 21 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 365 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2667 6 2039 3 4 204 2 6 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- - 5/2 F(n) F(n + 1) 22 22 319 753 3 15506 3 2762 3 132017 4 + --- F(n) - ----- F(n + 1) - ---- F(n) F(n + 1) + ------ F(n + 1) 319 319 319 638 14669 2 534 7 27960 3 + ----- F(n) F(n + 1) + --- F(n + 1) - ----- F(n) F(n + 1) 638 11 319 7 6263 2 10425 2 2 + 505 F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 638 319 201 2 5 3 5 8 + --- F(n) F(n + 1) - 435 F(n) F(n + 1) - 415/2 F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 366 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 6 G(n) = -1/12 F(n) (289 F(n) F(n + 1) - 200 F(n) F(n + 1) 5 2 6 + 222 F(n) F(n + 1) - 32 + 115 F(n + 1) - 108 F(n + 1) 2 4 3 + 174 F(n) F(n + 1) - 13 F(n) F(n + 1) + 2262 F(n) F(n + 1) 7 2 2 8 - 13300 F(n) F(n + 1) - 870 F(n) F(n + 1) + 5500 F(n + 1) 4 3 5 3 3 2 - 5481 F(n + 1) + 11800 F(n) F(n + 1) - 354 F(n) F(n + 1) - 4 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 367 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 2 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 9 30 2 7175 2 8 G(n) = -75/4 F(n) F(n + 1) + -- F(n) - ---- F(n) F(n + 1) 11 22 101 2 5 1871 3 3 141 3 4 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 11 11 22 39125 3 7 4 2 21 4 3 + ----- F(n) F(n + 1) + 207/2 F(n) F(n + 1) - -- F(n) F(n + 1) 44 22 35905 5 5 239 1733 5 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 44 11 44 29 6 18 2 7049 2 4 + -- F(n) F(n + 1) + -- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 11 22 3 3 51 2 10 - 8/11 F(n) - 5/22 F(n + 1) - -- F(n) F(n + 1) + 5/22 F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 368 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 2 15 2 G(n) = 7/22 F(n) - 5/22 F(n + 1) + 5/22 F(n + 1) + -- F(n) 22 15 4 5 2 3 - -- F(n) F(n + 1) + 6/11 F(n) F(n + 1) + 4/11 F(n) F(n + 1) 22 2 4 3 2 26 3 3 + 8/11 F(n) F(n + 1) - 7/22 F(n) F(n + 1) - -- F(n) F(n + 1) 11 15 4 67 4 2 5 - -- F(n) F(n + 1) + -- F(n) F(n + 1) - 4 F(n) F(n + 1) 11 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 369 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 353 2 784 3 15 3 393 6 1041 6 G(n) = --- F(n + 1) + --- F(n) - -- F(n + 1) - --- F(n + 1) + ---- F(n) 638 319 29 638 638 3041 3 3 1223 2 4 1481 2 5 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 638 2857 4 2 2584 6 1757 3 4 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 638 2072 5 2841 5 2 185 7 1333 7 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n + 1) - ---- F(n) 319 319 319 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 370 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 495 5 2 6 4702 3 G(n) = --- F(n) F(n + 1) + 185/6 F(n) F(n + 1) + ---- F(n) F(n + 1) 14 21 435 2 79 2 331 7871 2 2 + --- F(n + 1) + -- F(n) - --- - ---- F(n) F(n + 1) 28 84 84 84 3 5 2 4 7 + 1075 F(n) F(n + 1) + 45/7 F(n) F(n + 1) - 7615/6 F(n) F(n + 1) 107 43279 4 435 6 8 - --- F(n) F(n + 1) - ----- F(n + 1) - --- F(n + 1) + 3115/6 F(n + 1) 21 84 28 3 3 477 3 - 270/7 F(n) F(n + 1) + --- F(n) F(n + 1) 14 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 371 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 2 3 17 2 G(n) = -9/2 F(n) F(n + 1) - 5/22 F(n + 1) - 5/22 F(n) - -- F(n) 22 3 168 2 4 2 + 5/22 F(n + 1) - --- F(n) F(n + 1) - 19/2 F(n) F(n + 1) 11 2 4 41 465 4 3 + 9 F(n) F(n + 1) + -- F(n) F(n + 1) + --- F(n) F(n + 1) 11 11 245 3 4 365 2 5 81 2 - --- F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 22 175 6 + --- F(n) F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 372 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 4 181 4 776 3 7 G(n) = --- F(n + 1) - --- F(n) - --- F(n) - 75 F(n) F(n + 1) 319 319 319 17375 6 24366 3 4 4 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - 355/2 F(n) F(n + 1) 319 319 45960 4 3 11290 3 4 2 6 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + 355/2 F(n) F(n + 1) 319 319 39410 2 5 8710 2 2 10105 2 - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 638 3 5 17839 2 200 3 + 45/2 F(n) F(n + 1) - ----- F(n) F(n + 1) + --- F(n) F(n + 1) 319 29 20 3 - --- F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 373 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 15 4 13 7 4 3 3 4 G(n) = -- F(n) + -- F(n) F(n + 1) - 2 F(n) F(n + 1) - 21 F(n) F(n + 1) 22 11 977 5 3 19 2 79 2 + 5/22 - --- F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 11 22 11 2 2 356 2 6 87 3 + 65/2 F(n) F(n + 1) - --- F(n) F(n + 1) - -- F(n) F(n + 1) 11 11 757 3 5 5 2 12 3 3 + --- F(n) F(n + 1) + 39/2 F(n) F(n + 1) + -- F(n) - 5/22 F(n + 1) 11 11 2 5 657 4 4 + 13/2 F(n) F(n + 1) + --- F(n) F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 374 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 154999 9 8 291883 8 G(n) = ------- F(n + 1) - 5/2 F(n + 1) + ------ F(n) F(n + 1) 990 990 7 199847 2 3 273 2 2 + 15/2 F(n) F(n + 1) + ------ F(n) F(n + 1) + --- F(n) F(n + 1) 495 22 322879 8 474823 7 2 2 6 - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) - 35/4 F(n) F(n + 1) 990 990 227 3 10592 79021 303 203 4 + --- F(n) F(n + 1) + ----- F(n) + ----- F(n + 1) + --- - --- F(n + 1) 22 165 990 44 44 21559 9 8 417 3 7561 4 - ----- F(n) - 35/4 F(n) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 330 22 66 8467 5 + ---- F(n + 1) 110 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 375 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 19 10 73 68 181 2 G(n) = -- F(n + 1) - -- + -- F(n) F(n + 1) + --- F(n + 1) 12 24 15 60 427 5 347 5 3 5 5 + --- F(n) F(n + 1) - --- F(n) F(n + 1) - 1081/6 F(n) F(n + 1) 15 10 203 6 2 1925 6 4 7 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) + 41/6 F(n) F(n + 1) 12 12 7 3 4 6 67 4 + 765/2 F(n) F(n + 1) + 230/3 F(n) F(n + 1) + -- F(n + 1) 20 97 3 4 2 6 6 - -- F(n) F(n + 1) - 569/4 F(n) F(n + 1) - 23/5 F(n + 1) - 14/3 F(n) 15 89 8 37 8 217 4 4 + -- F(n) - --- F(n + 1) + --- F(n) F(n + 1) 24 120 24 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 376 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 76 6 63 5 57 2 G(n) = -- F(n) F(n + 1) - -- F(n) F(n + 1) - --- F(n) F(n + 1) 55 11 110 229 2 6 10 7 5 2 - --- F(n) F(n + 1) + 3/22 F(n) + -- F(n) + 21/2 F(n) F(n + 1) 110 11 6 52 225 2 4 + 9/110 F(n) F(n + 1) + -- F(n) F(n + 1) + --- F(n) F(n + 1) 11 22 183 4 3 135 5 2 21 2 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - 5/22 F(n + 1) + -- F(n) 22 11 22 3 + 5/22 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 377 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 4 4 6 2 G(n) = -1/4 F(n + 1) - 1 + 3/2 F(n + 1) - 181/6 F(n) - 60 F(n) F(n + 1) 7 5 6 + 245/3 F(n) F(n + 1) - 5/2 F(n) F(n + 1) + 2/3 F(n) 3 5 5 3 3 3 + 97/6 F(n) F(n + 1) - 67/2 F(n) F(n + 1) - 23/6 F(n) F(n + 1) 8 8 2 4 4 2 + 53/2 F(n) - 1/2 F(n + 1) + 3/2 F(n) F(n + 1) + 7/2 F(n) F(n + 1) 2 + 1/4 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 378 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5169 8 2558 2 7 330 3 3 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 3190 957 29 48490 8 2532 468 2 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) 957 319 319 8040 4 2 2280 5 8925 2 4 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 638 105 6 113 9 105 9 767 6 3 - --- F(n + 1) + --- F(n) + --- F(n + 1) - ---- F(n) F(n + 1) 638 66 638 4785 30790 7 2 49663 4 4593 2 3 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 957 9570 290 11015 + ----- F(n) 1914 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 379 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 375 3 4 1261 2 3 5 G(n) = ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + 45/2 F(n) F(n + 1) 11 22 1235 6 1410 2 5 30 3 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + 5/22 - -- F(n) 22 11 11 3 4 3235 4 3 3 - 5/22 F(n + 1) - 3/2 F(n) + ---- F(n) F(n + 1) + 153/2 F(n) F(n + 1) 22 7 183 2 2 2 - 75 F(n) F(n + 1) + --- F(n) F(n + 1) - 27 F(n) F(n + 1) 11 2 6 3 4 4 + 355/2 F(n) F(n + 1) + 7 F(n) F(n + 1) - 355/2 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 380 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 8 2 G(n) = -1/12 F(n) (-91 - 12774 F(n + 1) + 12850 F(n + 1) + 349 F(n + 1) 2 3 2 4 + 19 F(n) + 5398 F(n) F(n + 1) + 205 F(n) F(n + 1) 3 3 5 3 3 + 754 F(n) F(n + 1) + 27100 F(n) F(n + 1) - 885 F(n) F(n + 1) 2 6 2 2 7 + 150 F(n) F(n + 1) - 2156 F(n) F(n + 1) - 31250 F(n) F(n + 1) 5 6 + 775 F(n) F(n + 1) - 104 F(n) F(n + 1) - 340 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 381 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1879 9 535 8 859 4 661 15749 G(n) = ---- F(n) - --- F(n) - --- F(n + 1) + --- - ----- F(n) 205 33 22 33 2255 1531 14917 7 2 3343 7 + ---- F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 4510 410 33 2 2 7811 8 620 8 + 31/6 F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n + 1) 410 33 1024 3 3035 7 19383 6 3 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 11 33 205 3103 4 1396 8 1172 2 6 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 410 41 11 6663 5 6709 9 + ---- F(n + 1) - ---- F(n + 1) 410 410 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 382 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(2 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 G(n) = -1/6 F(n) (-2 F(n + 1) - 2 F(n + 1) + F(n) + 3 F(n) F(n + 1) 2 3 - 5 F(n) F(n + 1) + 5 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 383 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 G(n) = 5/22 F(n + 1) - 5/22 F(n + 1) - 1/2 F(n) + 7/11 F(n) F(n + 1) 2 25 2 3 37 3 2 12 4 - 6/11 F(n) + -- F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 22 22 11 45 5 + -- F(n) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 384 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 63 2 1802 7 83 7 2703 5 2 G(n) = ---- F(n + 1) + ---- F(n) - ---- F(n + 1) + ---- F(n) F(n + 1) 319 319 1276 638 21 4 2 1753 6 21 2 4 + -- F(n) F(n + 1) - ---- F(n) F(n + 1) - -- F(n) F(n + 1) 58 638 58 1861 2 5 782 3 293 3 126 3 3 + ---- F(n) F(n + 1) - --- F(n) + ---- F(n + 1) + --- F(n) F(n + 1) 1276 319 1276 319 7673 4 3 105 5 63 6 21 6 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) + --- F(n + 1) 1276 319 319 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 385 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 2 2 G(n) = 1/6 F(n) (3 + F(n + 1) - F(n) F(n + 1) - F(n) + 14 F(n) F(n + 1) 3 4 - 20 F(n) F(n + 1) + 4 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 386 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 10 5 45 3 3 61 4 2 87 2 G(n) = --- F(n) F(n + 1) + -- F(n) F(n + 1) + -- F(n) F(n + 1) + -- F(n) 11 11 22 22 2 23 2 12 2 15 3 + 5/22 F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) 22 11 22 3 43 5 73 6 - 5/22 F(n + 1) - -- F(n) F(n + 1) - -- F(n) 11 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 387 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 4 2 3 G(n) = 1/3 F(n) (-6 F(n) + 2 F(n + 1) + F(n) F(n + 1) - F(n) F(n + 1) 3 2 4 - 22 F(n) F(n + 1) + 26 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 388 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 29 6 148 3 4 G(n) = -F(n) - 5/22 - -- F(n) F(n + 1) - --- F(n) F(n + 1) 22 11 399 6 141 2 5 199 5 2 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 22 22 22 93 3 140 4 3 2 2 3 + -- F(n) + --- F(n) F(n + 1) - 5/2 F(n) F(n + 1) + 3 F(n) F(n + 1) 22 11 3 3 + 5/22 F(n + 1) + F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 389 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 2 5 G(n) = 97/2 F(n) F(n + 1) - 97/6 F(n) F(n + 1) - 22/3 F(n) F(n + 1) 6 3 4 3 - 851/3 F(n) F(n + 1) + 767/3 F(n) F(n + 1) - 1/6 F(n) + 19/6 F(n) 3 7 - 118 F(n + 1) + 118 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 390 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 24132 7 39 2 20 2 1315 3 G(n) = ----- F(n + 1) - --- F(n) + --- F(n + 1) + ---- F(n) 319 638 319 638 24152 3 793 2 5 79 - ----- F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 638 319 123 2 1881 2 101547 3 4 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) 11 58 638 117331 6 - ------ F(n) F(n + 1) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 391 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 4 G(n) = -1/84 F(n) (28 F(n) F(n + 1) + 665 F(n) F(n + 1) 5 2 6 7 + 1953 F(n) F(n + 1) - 1160 F(n) F(n + 1) - 26 F(n + 1) 4 3 7 3 - 1750 F(n) F(n + 1) + 250 F(n) + 2 F(n) - 4 F(n + 1) + 42 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 392 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 151 5 2 13 2 G(n) = -9/22 F(n) F(n + 1) + --- F(n) F(n + 1) + 5/22 F(n + 1) - -- F(n) 22 11 40 2 4 103 3 3 85 4 2 + -- F(n) F(n + 1) - --- F(n) F(n + 1) - -- F(n) F(n + 1) 11 22 22 3 3 2 12 2 - 5/22 F(n + 1) + 2/11 F(n) + 6/11 F(n) F(n + 1) - -- F(n) F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 393 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 745 3 5 300 4 3 167 7 G(n) = --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 58 319 319 347 2 30 7 1115 8 150 6 20 - --- F(n) F(n + 1) + --- F(n) - ---- F(n) + --- F(n) F(n + 1) - --- 638 319 319 319 319 1558 3 15523 4 4 38113 7 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 145 638 1595 2676 2 2 1213 2 6 148 3 20 3 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) + --- F(n + 1) 1595 3190 319 319 360 2 5 - --- F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 394 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 777 2 1717 7 4 G(n) = --- F(n) F(n + 1) + ---- F(n + 1) + 1/2 F(n + 1) - 8/11 22 22 3491 3 4 2109 6 3 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - F(n) F(n + 1) 22 11 139 2 2 2 179 2 5 - --- F(n) F(n + 1) - 5/2 F(n) F(n + 1) + --- F(n) F(n + 1) 11 22 3 856 3 30 3 + 3 F(n) F(n + 1) - --- F(n + 1) + -- F(n) 11 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 395 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 239 2 3 21 7 106 3 5 G(n) = ---- F(n) F(n + 1) + -- F(n) F(n + 1) - --- F(n) F(n + 1) 22 22 11 5 3 18 3 246 6 2 - 4/11 F(n) F(n + 1) - -- F(n) F(n + 1) - --- F(n) F(n + 1) 11 11 157 4 4 4 48 4 43 4 + --- F(n) F(n + 1) + 9/2 F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) 11 11 11 111 5 523 7 5 91 + --- F(n) + --- F(n) F(n + 1) - 5/22 F(n + 1) - -- F(n) 22 22 22 8 + 5/22 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 396 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 16 911 3 G(n) = 4/105 F(n + 1) + -- F(n) F(n + 1) + --- F(n) F(n + 1) 35 168 6 6 8 8 - 4/105 F(n + 1) - 4/21 F(n) - 11/3 F(n) - 1/14 F(n + 1) 16 5 49 3 5 7 - -- F(n) F(n + 1) - 1/7 - -- F(n) F(n + 1) + 65/8 F(n) F(n + 1) 35 24 6 2 4051 5 3 3 3 - 225/7 F(n) F(n + 1) + ---- F(n) F(n + 1) + 8/21 F(n) F(n + 1) 168 4 + 3/14 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 397 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3963 3897 2 95778 9 2517 G(n) = ----- F(n) - ----- F(n) + ----- F(n) + ---- F(n) F(n + 1) 418 12122 6061 6061 675205 4 886487 8 1234593 2 3 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - ------- F(n) F(n + 1) 24244 24244 60610 6207497 3 2 690 9 690 2 - ------- F(n) F(n + 1) + ---- F(n + 1) - ---- F(n + 1) 121220 6061 6061 2150301 2 7 82710 8 - ------- F(n) F(n + 1) + ----- F(n) F(n + 1) 30305 6061 13789617 7 2 + -------- F(n) F(n + 1) 121220 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 398 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2935 8 11725 4 24147 2 6 G(n) = ----- F(n + 1) + ----- F(n + 1) + ----- F(n) F(n + 1) 638 1276 1276 3 691 3 95 6 437 7 + 9/116 F(n + 1) - ---- F(n) - ---- F(n) F(n + 1) - ---- F(n) 2552 1276 2552 19 7 437 2 5 5935 1395 8 - ---- F(n + 1) + ---- F(n) F(n + 1) - ---- + ---- F(n) 1276 1276 1276 1276 1729 5 2 1765 5 3 14765 6 2 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 2552 319 319 8887 7 5766 3 1805 3 4 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 319 2552 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 399 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 4 G(n) = -1/84 F(n) (-499 + 131350 F(n) F(n + 1) - 62840 F(n + 1) 3 2 6 + 30 F(n) F(n + 1) + 27155 F(n) F(n + 1) + 3450 F(n) F(n + 1) 3 8 7 + 4136 F(n) F(n + 1) + 63325 F(n + 1) - 154775 F(n) F(n + 1) 2 2 2 2 - 11297 F(n) F(n + 1) - 5 F(n) - 30 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 400 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 543445 2 7 20 9 4 20 G(n) = ------ F(n) F(n + 1) - --- F(n + 1) + 7/22 F(n) + --- 319 319 319 96966 2 3 12 3 41427 3 2 - ----- F(n) F(n + 1) - -- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 11 319 15309 4 219716 4 3585 - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + ---- F(n) 319 319 638 19490 3 6 82415 4 5 3 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - 2/11 F(n) F(n + 1) 319 58 40245 8 2 2 - ----- F(n) F(n + 1) + F(n) F(n + 1) 58 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 401 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 90 63 2 3 238935 8 G(n) = - -- + --- F(n) F(n + 1) + 3/29 F(n) + ------ F(n + 1) 29 319 638 118425 4 105 3 481605 3 5 - ------ F(n + 1) - --- F(n + 1) + ------ F(n) F(n + 1) 319 638 638 94733 3 26215 7 45 2 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - --- F(n) F(n + 1) 638 29 319 42539 2 2 16469 3 13180 4 4 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 638 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 402 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 4 G(n) = 1/84 F(n) (-3124 F(n) F(n + 1) - 28125 F(n) F(n + 1) + 63325 F(n + 1) 4 2 2 2 2 + 527 F(n) - 23 F(n) - 12 F(n + 1) + 10770 F(n) F(n + 1) 7 8 + 154775 F(n) F(n + 1) + 12 F(n) F(n + 1) - 63325 F(n + 1) 3 5 2 6 - 131350 F(n) F(n + 1) - 3450 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 403 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6609 2 1496925 2 3643454 6 G(n) = ------ F(n) F(n + 1) - ------- F(n + 1) + ------- F(n + 1) 12122 497002 248501 489 3 402461 9 1350 2 - ---- F(n) + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 6061 17138 6061 58827229 6 4 33841062 7 3 - -------- F(n) F(n + 1) + -------- F(n) F(n + 1) 497002 248501 38602793 8 2 26080101 9 - -------- F(n) F(n + 1) + -------- F(n) F(n + 1) 497002 497002 2156765 55160 5 5733403 10 - ------- F(n) F(n + 1) - ----- F(n) F(n + 1) - ------- F(n + 1) 497002 22591 497002 690 3 54060 10 - ---- F(n + 1) - ----- F(n) 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 404 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 83419 9 5375 5 48 3 643 8 G(n) = ------ F(n) + ---- F(n + 1) + -- F(n) F(n + 1) + ---- F(n) F(n + 1) 5742 5742 55 2871 21359 5 4 57 5 3 21 4 8 - ----- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n + 1) - 6/11 F(n) 5742 55 55 8 119599 5 2215 4135 4 - 3/110 F(n + 1) + ------ F(n) - ---- F(n + 1) - ---- F(n) F(n + 1) 5742 2871 1914 6455 4 5 6 2 15 7 - ---- F(n) F(n + 1) - 3/22 F(n) F(n + 1) - -- F(n) F(n + 1) 1914 11 64894 7 2 62153 8 78 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + --- 957 957 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 405 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 3 5 7 3 G(n) = -102/7 F(n) F(n + 1) - 3 F(n) F(n + 1) + 985/3 F(n) F(n + 1) 7 6 4 6 2 - 3/2 F(n) F(n + 1) - 1550/7 F(n) F(n + 1) - 7/4 F(n) F(n + 1) 821 5 5 5 3 8107 10 8 + --- F(n) F(n + 1) + 7/2 F(n) F(n + 1) - ---- F(n) - 1/12 F(n) 21 84 6 2 6 8 + 527/6 F(n) - 4/7 F(n) + 2/21 F(n + 1) + 1/6 F(n + 1) + 1/3 4 2 31 4 4 - 1/2 F(n + 1) - 2/21 F(n + 1) + -- F(n) F(n + 1) 12 1755 9 3 3 - ---- F(n) F(n + 1) + 24/7 F(n) F(n + 1) 14 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 406 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(4 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 111429 2 3000 2 776131 6 330625 10 G(n) = ------ F(n + 1) - ---- F(n) - ------ F(n + 1) + ------ F(n + 1) 319 319 1276 1276 21 2 3 101825 9 5 + -- F(n) F(n + 1) - ------ F(n) F(n + 1) + 7213/2 F(n) F(n + 1) 44 29 53959 4 488431 4 2 + ----- F(n) F(n + 1) - 9/4 F(n) F(n + 1) - ------ F(n) F(n + 1) 638 638 290005 2 4 9143225 2 8 5 - ------ F(n) F(n + 1) + ------- F(n) F(n + 1) - 3/4 F(n + 1) 319 1276 27 4 7223675 4 6 129 747 + -- F(n) F(n + 1) - ------- F(n) F(n + 1) + --- F(n) + ---- F(n + 1) 22 1276 319 1276 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 407 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 5 4 G(n) = 1/6 F(n) (-11 F(n) + 3 F(n) + 2 F(n) + 4 F(n + 1) + F(n) F(n + 1) 3 2 4 2 - 46 F(n) F(n + 1) + 53 F(n) F(n + 1) - 6 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 408 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(n + j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 49 3 392 6 G(n) = - 5/22 + 5/22 F(n + 1) + -- F(n) - --- F(n) F(n + 1) 22 11 948 2 5 1717 4 3 3 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) + 1/2 F(n) F(n + 1) 11 22 267 2 2 2 3 - --- F(n) F(n + 1) - 2 F(n) F(n + 1) + 3/2 F(n) F(n + 1) 22 57 3 4 383 2 + -- F(n) F(n + 1) + --- F(n) F(n + 1) 22 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 409 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(n + j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 6 G(n) = 93/2 F(n) F(n + 1) - 44/3 F(n) F(n + 1) - 824/3 F(n) F(n + 1) 3 4 2 5 2 3 + 1501/6 F(n) F(n + 1) - 28/3 F(n) F(n + 1) + 2 F(n) F(n + 1) 3 4 3 2 5 + 8/3 F(n) + 13/2 F(n) F(n + 1) - 17/2 F(n) F(n + 1) - 3 F(n + 1) 3 7 - 2/3 F(n) + 3 F(n + 1) - 229/2 F(n + 1) + 229/2 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 410 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 160 2 6 3 2 4 G(n) = --- F(n) F(n + 1) + 4 F(n) F(n + 1) - 3 F(n) F(n + 1) 11 541 6 2 160 7 4 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + 1/2 F(n) F(n + 1) 22 11 2 3 40 8 - 5/2 F(n) F(n + 1) + 5/22 + 9/22 F(n) - 5/22 F(n + 1) - -- F(n) 11 131 3 74 7 81 2 2 + --- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 22 35 5 3 - -- F(n) F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 411 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 235 8 167 2 2 6 G(n) = 65/6 F(n + 1) - --- F(n) - --- F(n) - 13/4 F(n + 1) + 13/4 F(n + 1) 18 60 79 6 89 5 6 2 + -- F(n) - -- F(n) F(n + 1) - 451/9 F(n) F(n + 1) 20 10 7 2 4 31 - 155/9 F(n) F(n + 1) + 9/2 F(n) F(n + 1) + -- F(n) F(n + 1) 15 5 3 3 421 7 + 760/9 F(n) F(n + 1) + 41/2 F(n) F(n + 1) - --- F(n) F(n + 1) 18 355 4 - --- F(n + 1) + 80/9 18 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 412 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(n + 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 10 8 2725633 9 206264 9 10 8 G(n) = --- F(n + 1) + ------- F(n + 1) + ------ F(n) - -- F(n) 77 28710 4785 77 4 19 7 464051 8 24 + 2/11 F(n + 1) - -- F(n) F(n + 1) - ------ F(n) F(n + 1) + ---- 77 2610 2233 12 2 6 69644 4 73717 5 - -- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n + 1) 77 957 1595 176984 1400527 25 7 23 3 - ------ F(n) - ------- F(n + 1) + -- F(n) F(n + 1) + --- F(n) F(n + 1) 4785 28710 77 154 3452879 2 3 61 2 2 4944343 8 - ------- F(n) F(n + 1) - --- F(n) F(n + 1) + ------- F(n) F(n + 1) 14355 154 28710 4496333 7 2 + ------- F(n) F(n + 1) 14355 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 413 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(n + 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 7365 8 54 1664 3 7250 3 5 G(n) = ---- F(n + 1) - -- + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 22 11 11 11 2 3 1635 2 2 9145 7 - 9/2 F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 22 11 3626 4 364 3 680 2 6 - ---- F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 11 11 11 4 5 3 2 10 - 15/2 F(n) F(n + 1) + 7/2 F(n + 1) + 23/2 F(n) F(n + 1) + -- F(n) 11 41 - -- F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 414 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2415 3 6 7652 4 15485 8 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 22 11 22 2 3 19160 2 7 1579 3 2 - 633/2 F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 11 3 3 630 4 15410 4 5 + 12 F(n) F(n + 1) - --- F(n) F(n + 1) - ----- F(n) F(n + 1) 11 11 75 5 2 4 157 + -- F(n) F(n + 1) - 13/2 F(n) F(n + 1) + 11 F(n) F(n + 1) + --- F(n) 11 22 4 2 2 25 2 9 - 22 F(n) F(n + 1) - 5/22 F(n + 1) - -- F(n) + 5/22 F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 415 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 10 3 7 G(n) = - 1/6 + 5/12 F(n) - 37/4 F(n) + 32/7 F(n) F(n + 1) 2 2 6 4 1289 4 2 - 2/3 F(n) F(n + 1) - 976/7 F(n) F(n + 1) - ---- F(n) F(n + 1) 42 8 2 2017 9 2 - 510/7 F(n) F(n + 1) + ---- F(n) F(n + 1) - 2/21 F(n + 1) 42 6 4051 7 3 2 4 + 2/21 F(n + 1) + ---- F(n) F(n + 1) - 4/7 F(n) F(n + 1) 21 317 4 6 4 + --- F(n) F(n + 1) + 1/6 F(n + 1) 42 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 416 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(n + 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 95 2 8 240 9 G(n) = --- F(n + 1) - 7/957 F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 1755 3 3 5351 4 2 3481 5 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 638 638 788 5 4 1310 6 3 1812 6 4 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 957 957 29 823 7 2 88523 7 3 1226 8 - --- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 957 638 957 63451 8 2 2357 9 115 6 216 9 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n + 1) + --- F(n) 638 58 319 319 37 73 3 2 263 4 23 5 + --- F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n + 1) 957 319 638 957 3087 10 - ---- F(n) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 417 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 8 503 5 3 4 G(n) = -1/2 F(n + 1) - 11/3 F(n) + --- F(n) F(n + 1) - 1 + 3/2 F(n + 1) 12 6 6 6 2 2 + 1/30 F(n + 1) + 2/3 F(n) - 60 F(n) F(n + 1) - 1/30 F(n + 1) 5 7 - 13/5 F(n) F(n + 1) - 2/5 F(n) F(n + 1) + 25/4 F(n) F(n + 1) 13 3 5 4 2 3 3 + -- F(n) F(n + 1) + 6 F(n) F(n + 1) - 13/3 F(n) F(n + 1) 12 181 3 + --- F(n) F(n + 1) 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 418 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(2 n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 80 3 3 2 35 203 9 G(n) = -- F(n) F(n + 1) - 5/22 F(n + 1) - -- F(n) + --- F(n) 11 22 22 128 6 3 230 5 4 112 2 7 - --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 11 11 55 29 2 4 18 6 9 + 9/11 F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) + 5/22 F(n + 1) 11 11 6673 7 2 235 8 2 3 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) - 9/10 F(n) F(n + 1) 110 11 1673 3 2 173 4 2 41 5 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) F(n + 1) 110 22 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 419 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(2 n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 5 G(n) = -1/12 F(n) (752 F(n) F(n + 1) + 27100 F(n) F(n + 1) 4 2 3 3 6 + 375 F(n) F(n + 1) - 120 F(n) F(n + 1) + 45 F(n + 1) 7 2 6 2 - 31250 F(n) F(n + 1) + 150 F(n) F(n + 1) - 39 F(n + 1) 4 8 - 12776 F(n + 1) + 12850 F(n + 1) - 96 F(n) F(n + 1) 3 2 2 2 4 + 5402 F(n) F(n + 1) - 2156 F(n) F(n + 1) - 165 F(n) F(n + 1) 2 + 16 F(n) - 88) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 420 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n - j) F(2 n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 1295 3 45 3 243 2 63325 10 G(n) = ----- F(n + 1) + -- F(n) - --- F(n) + ----- F(n + 1) 11 22 22 11 10431 2 119375 3 7 3 4 - ----- F(n + 1) + ------ F(n) F(n + 1) + 525/2 F(n) F(n + 1) 22 11 18666 3 3 2 4 503 2 + ----- F(n) F(n + 1) - 1732 F(n) F(n + 1) + --- F(n) F(n + 1) 11 11 21141 5 6 159325 9 + ----- F(n) F(n + 1) - 280 F(n) F(n + 1) - ------ F(n) F(n + 1) 11 11 153 2 7 2183 - --- F(n) F(n + 1) + 235/2 F(n + 1) + ---- F(n) F(n + 1) 11 22 2 5 36475 2 8 58107 6 - 35/2 F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n + 1) 22 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 421 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n - j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 61 7 5 7 G(n) = -- F(n) F(n + 1) + 181/5 F(n) F(n + 1) + 215/9 F(n) F(n + 1) 45 6 4 2147 6 2 1159 5 5 - 667/6 F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 45 30 2 6 10 - 77/9 + 53/5 F(n + 1) - 23/2 F(n) - 53/5 F(n + 1) 8 2 1447 5 287 3 - 946/3 F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 15 15 154 2 2 7 3 199 8 4 - --- F(n) F(n + 1) + 785/3 F(n) F(n + 1) + --- F(n) + 77/9 F(n + 1) 15 18 5 3 113 9 - 514/9 F(n) F(n + 1) - --- F(n) F(n + 1) + 113/6 F(n) F(n + 1) 10 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 422 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(3 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 5 G(n) = -1/15 F(n) F(n + 1) (-F(n + 1) + F(n)) (8 F(n + 1) - 24 F(n + 1) F(n) 4 2 3 3 2 4 + 7 F(n + 1) F(n) + 26 F(n + 1) F(n) + 13 F(n + 1) F(n) 5 6 - 30 F(n + 1) F(n) + 4 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 423 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(3 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 14987 5 61117 6 3 3502 5 G(n) = ------ F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 24244 12122 6061 150093 2 3 8755 2 4 8755 4 2 + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 60610 6061 6061 123298 7 2 13745 8 10125 8 - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 30305 24244 12122 202721 2 7 227729 3 2 19267 - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) + ----- F(n + 1) 121220 60610 24244 2502 2 2821 2 1751 6 2502 9 - ---- F(n) - ---- F(n + 1) + ---- F(n + 1) + ---- F(n) 6061 6061 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 424 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(3 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2351 8 29 5 20 6 3 G(n) = ---- F(n) F(n + 1) + -- F(n + 1) - -- F(n) F(n + 1) - 8/21 F(n + 1) 420 12 21 3 2 9 7 + 8/35 F(n) + 4/35 F(n) F(n + 1) - 4/21 F(n) - 4/105 F(n) 7 29 133 6 3 601 7 2 + 8/21 F(n + 1) - -- F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 12 30 42 8 2 5 2 7 + 5/12 F(n) F(n + 1) + 8/105 F(n) F(n + 1) - 76/7 F(n) F(n + 1) 16 3 4 583 4 251 2 3 + -- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 21 60 60 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 425 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 11346 8 113904 3 3 182209 2 4 G(n) = ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 41591 207955 415910 7320 9 1830 9 9543 10 1830 10 + ----- F(n) + ----- F(n + 1) - ----- F(n) - ----- F(n + 1) 41591 41591 41591 41591 140658 2 8 25620 2 7 134856 4 6 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 207955 41591 41591 5490 4 5 670804 7 3 29280 5 4 - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 41591 207955 41591 48973 9 17568 3 2 19764 4 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 41591 41591 41591 5380 3 7 46848 3 6 3484 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 2189 41591 41591 117 155761 8 2 + ---- F(n) + ------ F(n) F(n + 1) 2189 83182 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 426 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 9 4 G(n) = 1/3135 F(n) (-7386 F(n + 1) + 4410 F(n + 1) + 18816 F(n) F(n + 1) 8 2 3 2 7 - 21985 F(n) F(n + 1) + 2726 F(n) F(n + 1) + 26952 F(n) F(n + 1) 7 2 9 - 28181 F(n) F(n + 1) - 535 F(n) + 535 F(n) + 4648 F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 427 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 38814 4 2 9782 4 3 372525 3 7 G(n) = ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + ------ F(n) F(n + 1) 551 6061 2204 4891 7 4891 2 28900 4 6 29346 3 4 + ---- F(n) - ---- F(n) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 6061 551 6061 33586 3 3 1070 3 937 5 - ----- F(n) F(n + 1) - ---- F(n + 1) + --- F(n) F(n + 1) 551 6061 29 29346 5 2 346825 5 5 189268 - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 6061 2204 6061 9506 2 131761 5 4891 6 + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 2204 6061 64225 9 2595 2 1070 2 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n + 1) 2204 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 428 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 6 6 3 G(n) = 1/420 F(n) (-240 F(n + 1) - 80 F(n) F(n + 1) - 1020 F(n) F(n + 1) 7 2 2 5 7 - 135 F(n) F(n + 1) + 160 F(n) F(n + 1) + 80 F(n + 1) 4 5 7 5 + 8370 F(n) F(n + 1) - 80 F(n) + 1134 F(n) - 1054 F(n) 2 8 + 320 F(n) F(n + 1) + 3270 F(n) F(n + 1) - 232 F(n + 1) 6 5 2 5 4 - 240 F(n) F(n + 1) + 160 F(n) F(n + 1) - 11025 F(n) F(n + 1) 9 5 - 180 F(n + 1) + 792 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 429 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1037 2 4 6117 3 3 17053 4 2 G(n) = ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 12122 6061 12122 3372 5 4798 5 2303 2 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 6061 6061 801 2 2276 2 1070 3 2276 3 + ---- F(n) F(n + 1) + ---- F(n) + ---- F(n + 1) - ---- F(n) 6061 6061 6061 6061 1070 6 - ---- F(n + 1) 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 430 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 16 6 608 5 192 16 2 G(n) = ---- F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n + 1) 385 385 385 385 4 116 4 16 2 2 - 4/77 F(n + 1) + --- F(n) + 4/77 + -- F(n) F(n + 1) 231 77 320 4 2 292 3 736 3 3 - --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 77 231 231 128 6 - --- F(n) 231 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 431 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 11097 2 1830 6 1830 5 22524 5 G(n) = ----- F(n) + ----- F(n + 1) - ----- F(n + 1) - ----- F(n) 7562 41591 41591 41591 77019 6 16605 4 2 188145 5 - ----- F(n) + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 83182 3781 41591 12691 4 6984 5 1674 2 3 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 41591 41591 41591 63110 3 2 29940 3 3 46041 4 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 41591 41591 41591 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 432 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 11127 6 31461 4 3 6371 2 G(n) = ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 24244 6061 24244 22893 5 2 120 3 545 3 1070 3 - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) - ---- F(n + 1) 24244 6061 638 6061 14681 7 9952 3 2703 3 4 - ----- F(n) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 12122 6061 836 3812 2 2 1093 4 1070 + ---- F(n) F(n + 1) + ---- F(n) + ---- 6061 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 433 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 5 G(n) = -1/231 F(n) (472 F(n) F(n + 1) + 1220 F(n) F(n + 1) 3 3 3 6 2 + 1570 F(n) F(n + 1) - 464 F(n) F(n + 1) + 40 F(n + 1) - 120 F(n + 1) 2 4 3 4 - 985 F(n) F(n + 1) - 16 F(n) F(n + 1) + 8 F(n + 1) 4 2 2 4 - 1725 F(n) F(n + 1) - 228 F(n) + 228 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 434 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4439 1070 1070 1915 4 4 G(n) = ----- F(n) + ---- F(n + 1) - ---- + ---- F(n) F(n + 1) 6061 6061 6061 22 229659 3 415 7 95 3 5 - ------ F(n) F(n + 1) + --- F(n) F(n + 1) - -- F(n) F(n + 1) 6061 11 11 5509 4 24171 3 2 18471 4 + ---- F(n) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 6061 6061 4362 4 83359 2 2 2098 2 3 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 6061 6061 1945 2 6 19389 3 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 22 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 435 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(3 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 3 G(n) = -1/84 F(n) (26 F(n) - 4096 F(n + 1) + 1667 F(n) F(n + 1) 2 3 7 - 5 F(n + 1) + 209 F(n) F(n + 1) - 26 - 9975 F(n) F(n + 1) 8 2 6 + 5 F(n) F(n + 1) + 4125 F(n + 1) - 150 F(n) F(n + 1) 2 2 3 5 - 630 F(n) F(n + 1) + 8850 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 436 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 n - 2 j) F(3 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 340 3 10360 2 3 650 2 2 G(n) = --- F(n) F(n + 1) + ----- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 11175 8 100450 3 6 6225 2 7 + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 29 319 319 254 49850 9 19818 4 6436 3 2 + --- - ----- F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 319 1092 3 234 4 48349 5 254 - ---- F(n) F(n + 1) - --- F(n + 1) + ----- F(n + 1) - --- F(n) 319 319 319 319 1481 + ---- F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 437 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 n - 2 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4064 2 22866328 2 14791577 7 3 G(n) = ---- F(n) F(n + 1) - -------- F(n + 1) - -------- F(n) F(n + 1) 6061 4697275 939455 95279 8 2 10885172 9 43077887 6 + ----- F(n) F(n + 1) + -------- F(n) F(n + 1) + -------- F(n + 1) 6061 939455 9394550 1070 3 1591163 4792733 5 - ---- F(n + 1) + ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 6061 4697275 427025 115804 9 4313269 10 68 3 15241737 2 - ------ F(n) F(n + 1) + ------- F(n + 1) - ---- F(n) - -------- F(n) 187891 9394550 6061 4697275 15294437 10 4432 2 + -------- F(n) - ---- F(n) F(n + 1) 4697275 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 438 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 n - 2 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 6 6 3 G(n) = -1/84 F(n) (4 F(n + 1) - 6 F(n) F(n + 1) - 906 F(n) F(n + 1) 6 5 4 5 - 72 F(n) F(n + 1) + 693 F(n) F(n + 1) - 6 F(n + 1) 4 3 4 3 6 + 60 F(n) F(n + 1) - 256 F(n) F(n + 1) + 132 F(n) F(n + 1) 2 8 2 3 + 52 F(n) F(n + 1) + 15 F(n) F(n + 1) - 178 F(n) F(n + 1) 7 2 4 5 5 3 + 1182 F(n) F(n + 1) - 714 F(n) F(n + 1) + 19 F(n) + 11 F(n) 7 - 30 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 439 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 n - 2 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 911 911 2 201607 5 2796 G(n) = ---- F(n) + --- F(n) - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 638 638 638 319 40 3 6 20 10 20 4 5 + --- F(n) F(n + 1) - --- F(n + 1) + --- F(n) F(n + 1) 319 319 319 212405 4 6 40 8 103890 9 + ------ F(n) F(n + 1) - --- F(n) F(n + 1) + ------ F(n) F(n + 1) 319 319 319 224 2 3 3689 2 4 20 2 7 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 319 29 319 254880 2 8 1742 3 2 20172 3 3 - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 319 9410 3 7 1986 4 24239 4 2 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 638 20 9 228 4 + --- F(n + 1) + --- F(n) F(n + 1) 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 440 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(3 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 655 6 20 6 129 3 1171 2 20 3 G(n) = ----- F(n) - --- F(n + 1) - --- F(n) + ---- F(n) + --- F(n + 1) 1276 319 319 1276 319 101 2 21 2 3445 4 2 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 638 1276 845 5 135 5 175 2 4 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 1276 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 441 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 n - 2 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 2 4 G(n) = 1/84 F(n) (30 F(n + 1) + 48 F(n) F(n + 1) - 35 F(n) F(n + 1) 2 2 3 3 2 - 96 F(n) F(n + 1) + 120 F(n) F(n + 1) - 265 F(n) F(n + 1) 4 3 3 + 230 F(n) F(n + 1) - 32 F(n + 1) - 40 F(n) + 40 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 442 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 n - 2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1070 1070 2 2802 2 2802 5 G(n) = ----- F(n + 1) + ---- F(n + 1) + ---- F(n) - ---- F(n) 6061 6061 6061 6061 1816 4 745 5 3618 2 3 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 6061 6061 15775 2 4 6478 3 2 7400 3 3 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 12122 6061 6061 8142 4 42305 4 2 18615 5 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 12122 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 443 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 n - 2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 305 3 127 3 459 4 1068 4 3 G(n) = - 7/29 - --- F(n) - --- F(n + 1) + --- F(n) + ---- F(n) F(n + 1) 638 638 638 319 919 5 2 89 2 5 993 3 4 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 638 638 52 6 82 3 24 3 + -- F(n) F(n + 1) - --- F(n) F(n + 1) - -- F(n) F(n + 1) 29 319 29 97 4 7 + --- F(n + 1) + 3/22 F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 444 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 n - 2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4105 7 6 9175 3 4 G(n) = ---- F(n + 1) - 815/7 F(n) F(n + 1) + ---- F(n) F(n + 1) 84 84 164 4 79 2 62 2 3 + --- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 21 14 21 265 2 5 226 3 2 73 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - -- F(n) + 26/7 F(n + 1) 42 21 84 4105 3 73 3 5 1609 2 - ---- F(n + 1) + -- F(n) - 26/7 F(n + 1) + ---- F(n) F(n + 1) 84 84 84 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 445 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 n - 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 58750 6 1051 2 332 2 4 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 551 209 209 4240 2 5 54805 3 4 5722 3 3 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 551 551 209 5696 5 104702 2 4168 3 + ---- F(n) F(n + 1) - ------ F(n) F(n + 1) - ---- F(n) 209 6061 6061 24686 543505 3 49215 7 - ----- F(n) F(n + 1) + ------ F(n + 1) - ----- F(n + 1) 6061 12122 1102 2444 6 334 2 4168 2 - ---- F(n + 1) + --- F(n + 1) + ---- F(n) 209 29 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 446 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 n - 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 563 5 603 4 259 4 G(n) = - --- - --- F(n) + --- F(n) - --- F(n) F(n + 1) 319 638 638 638 23857 3 4273 2 2 415 7 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 638 319 11 1945 2 6 2451 3 1915 4 4 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 22 638 22 95 3 5 243 2 3 805 4 - -- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 11 319 638 500 3 2 20 - --- F(n) F(n + 1) + --- F(n + 1) 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 447 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 n - 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 3 3 2 G(n) = -1/84 F(n) (-48 F(n + 1) + 384 F(n) F(n + 1) - 451 F(n) F(n + 1) 2 5 5 4 - 810 F(n) F(n + 1) - 113 F(n) + 50 F(n + 1) - 125 F(n) F(n + 1) 6 3 4 4 + 140 F(n) F(n + 1) - 560 F(n) F(n + 1) + 368 F(n) F(n + 1) 4 3 5 2 2 - 1960 F(n) F(n + 1) + 3080 F(n) F(n + 1) - 68 F(n) F(n + 1) 3 + 113 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 448 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 n - 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 1143 4 13 3 6 1442 4 2 G(n) = ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 1276 6380 319 9026 6 3 20055 7 2 9561 8 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 1595 17256 4 434 6 159 181 5 + ----- F(n) F(n + 1) + --- F(n) - ---- F(n) + ---- F(n) F(n + 1) 1595 319 1276 1276 4857 3 3 3339 5 1577 9 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) 1276 1276 1276 937 20 2 20 9 - ---- F(n) F(n + 1) + --- F(n + 1) - --- F(n + 1) 1276 319 319 279 3 2 - ---- F(n) F(n + 1) 3190 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 449 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 n - j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 243 2 111191 2 1367 G(n) = ----- F(n) F(n + 1) + ------ F(n + 1) - ----- F(n) F(n + 1) 12122 12122 12122 282600 3 7 415649 2 4 7515 2 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 551 6061 12122 666143 5 359675 9 550223 3 3 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 6061 551 12122 6153 2 690 3 6153 3 1534732 6 - ----- F(n) - ---- F(n + 1) + ----- F(n) + ------- F(n + 1) 12122 6061 12122 6061 28900 2 8 289025 10 - ----- F(n) F(n + 1) - ------ F(n + 1) 551 1102 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 450 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 n - j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 19 8 447337 2 455737 10 19 8 G(n) = --- F(n + 1) + ------ F(n) - ------ F(n) + --- F(n) 112 58800 58800 112 6387 9 676091 8 2 361 6 2 - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) - --- F(n) F(n + 1) 196 11760 560 137981 9 209 7 296223 5 + ------ F(n) F(n + 1) - --- F(n) F(n + 1) + ------ F(n) F(n + 1) 5880 280 9800 39 3 51 2 2 378311 2 4 + --- F(n) F(n + 1) + --- F(n) F(n + 1) + ------ F(n) F(n + 1) 280 560 11760 95 2 6 66163 4 + --- F(n) F(n + 1) - 3/112 + ----- F(n) F(n + 1) - 1/7 F(n + 1) 112 14700 942073 2 747127 10 32491 6 + ------ F(n + 1) - ------ F(n + 1) - ----- F(n + 1) 58800 58800 9800 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 451 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 n - j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 59 20 14 5 42475 4 6 621 2 G(n) = --- F(n) + --- F(n + 1) + --- F(n) - ----- F(n) F(n + 1) + --- F(n) 58 319 319 319 638 105977 4 2 20797 5 136850 3 7 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 638 319 319 3764 3 3 176 4 1680 3 2 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 29 29 319 23550 9 127440 5 5 94287 5 - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 638 23274 108 4 20 10 - ----- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n + 1) 319 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 452 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(3 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 145 2 4 145 6 2 G(n) = ---- F(n + 1) - 2/3 F(n + 1) + --- F(n + 1) - 2/3 F(n) 12 12 3 2 4 2 2 - 8/3 F(n) F(n + 1) - 5 F(n) F(n + 1) + 2/3 F(n) F(n + 1) 3 5 + 4/3 F(n) F(n + 1) + 23/6 F(n) F(n + 1) + 2/3 - 55/2 F(n) F(n + 1) 3 3 + 30 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 453 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 245 5 17 4 2415 4 2 G(n) = ---- F(n) F(n + 1) + -- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 22 638 679 5 103 4 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - 3/2 F(n) F(n + 1) 319 638 709 331 2 21 5 371 6 687 - ---- F(n + 1) - --- F(n + 1) + -- F(n + 1) + --- F(n + 1) + ---- F(n) 1595 638 55 638 1595 141 2 48 5 + --- F(n) - -- F(n) 319 55 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 454 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 16538 4 3 7560 5 160 5 2 G(n) = ------ F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 6061 551 319 190808 6 6231 34109 6 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 30305 12122 30305 119021 2 12915 4 2 2373 6 - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) 60610 1102 1102 882 5 8693 2 690 3 46425 2 - --- F(n) F(n + 1) - ----- F(n) F(n + 1) + ---- F(n + 1) + ----- F(n) 551 60610 6061 12122 10161 7 690 2 - ----- F(n) - ---- F(n + 1) 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 455 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 565 7 2 G(n) = 26/3 F(n) F(n + 1) + --- F(n + 1) - 31/6 F(n) F(n + 1) - 3/4 F(n) 12 6 3 4 + 4 F(n + 1) - 335/3 F(n) F(n + 1) + 425/4 F(n) F(n + 1) 217 2 2 3 2 5 + --- F(n) F(n + 1) + 8/3 F(n) F(n + 1) - 15/2 F(n) F(n + 1) 12 3 2 3 5 565 3 - 34/3 F(n) F(n + 1) + 3/4 F(n) - 4 F(n + 1) - --- F(n + 1) 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 456 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 n - j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 11179 2 577 5 677 3 14620 7 G(n) = ------ F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) - ----- F(n + 1) 638 22 638 319 32315 3 4 2681 69945 6 - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 638 638 113 2 39 2 4 589 3 3 + --- F(n) F(n + 1) + -- F(n) F(n + 1) - --- F(n) F(n + 1) 22 22 22 14640 3 677 2 3576 2 124 6 + ----- F(n + 1) + --- F(n) + ---- F(n + 1) - --- F(n + 1) 319 638 319 11 4485 2 5 + ---- F(n) F(n + 1) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 457 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 n - j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 101 6 17 6 6 2 G(n) = F(n + 1) + --- F(n + 1) + -- F(n) - 95/3 F(n) F(n + 1) 60 12 7 5 19 + 275/6 F(n) F(n + 1) - 25/6 F(n) F(n + 1) + -- F(n) F(n + 1) 30 4 2 53 5 217 4 8 + 6 F(n) F(n + 1) - 2/3 - -- F(n) F(n + 1) - --- F(n) + 52/3 F(n) 10 12 8 3 5 5 3 - 1/3 F(n + 1) + 67/6 F(n) F(n + 1) - 139/6 F(n) F(n + 1) 101 2 - --- F(n + 1) 60 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 458 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 3 3247 4 959 3 1887 3 4 20 G(n) = ---- F(n + 1) - ---- F(n) - --- F(n) - ---- F(n) F(n + 1) + --- 319 638 638 638 319 2083 8 845 2 6 79 4 3 + ---- F(n) + --- F(n) F(n + 1) + -- F(n) F(n + 1) 319 319 58 2029 5 2 111 3 2666 3 5 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 29 319 7880 5 3 2538 6 469 2 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 638 503 3 579 2 5 4 4 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + 55/2 F(n) F(n + 1) 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 459 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 n - j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 4 4 G(n) = 1/60 F(n) (585 F(n + 1) - 1200 F(n) F(n + 1) + 2233 F(n + 1) 3 7 7 - 2529 F(n) F(n + 1) + 2575 F(n) F(n + 1) + 3475 F(n) F(n + 1) 6 2 5 3 5 + 4010 F(n) F(n + 1) - 8636 F(n) F(n + 1) + 1680 F(n) F(n + 1) 8 6 8 2 6 - 1128 F(n + 1) - 640 F(n) + 1190 F(n) + 510 F(n) - 600 F(n + 1) - 465 F(n) F(n + 1) - 1060) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 460 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 9875 4 3 8293 5 6 25 6 G(n) = ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - -- F(n) F(n + 1) 792 396 33 515 6 3 2785 7 4 515 5 4 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 792 99 396 93 5 2 8093 4 7 515 4 + -- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 22 792 792 515 4 5 515 3 6 29 3 8 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - -- F(n) F(n + 1) 792 396 12 77 2 9 515 2 7 11 7 + -- F(n) F(n + 1) - --- F(n) F(n + 1) - -- F(n + 1) 72 792 72 5593 6 5 103 5 103 7 103 - ---- F(n) F(n + 1) + --- F(n) - --- F(n) - --- F(n + 1) 792 396 396 792 103 5 11 3 + --- F(n + 1) + -- F(n + 1) 792 72 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 461 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 4 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2852395 3 5451 3 1597375 11 G(n) = ------- F(n + 1) + ----- F(n) - ------- F(n + 1) 83182 83182 3781 10089705 3 4 8005625 10 - -------- F(n) F(n + 1) + ------- F(n) F(n + 1) 83182 7562 5727495 6 620275 2 10707 - ------- F(n) F(n + 1) - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 41591 83182 41591 27803 2 3053250 3 8 267400 2 5 + ----- F(n) F(n + 1) - ------- F(n) F(n + 1) + ------ F(n) F(n + 1) 41591 3781 2189 814875 2 9 5451 2 32286195 7 - ------ F(n) F(n + 1) - ----- F(n) + -------- F(n + 1) 7562 83182 83182 1830 2 + ----- F(n + 1) 41591 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 462 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 4 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 10236 3 1879 2 3016125 10 G(n) = ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ------- F(n) F(n + 1) 6061 12122 1102 2300625 3 8 335835 3 4 1011345 3 - ------- F(n) F(n + 1) - ------ F(n) F(n + 1) + ------- F(n + 1) 1102 1102 12122 93354 2 408075 6 8409 4 9789 - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) + ----- F(n + 1) - ----- 6061 1102 12122 12122 153500 2 9 10713 2 2 37143 3 - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 551 6061 12122 351755 2 5 9789 3 555905 7 + ------ F(n) F(n + 1) + ----- F(n) + ------ F(n + 1) 1102 12122 551 1203625 11 - ------- F(n + 1) 1102 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 463 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20325 3 4 25325 5 2 5000 2 5 G(n) = ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 12122 6061 6061 4773 3 503 2 1137 - ----- F(n) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 12122 6061 6061 5634 2 30325 4 3 690 3 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n + 1) 6061 12122 6061 690 2 4773 2 - ---- F(n + 1) + ----- F(n) 6061 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 464 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 4 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 643075 6 1668225 4 3 G(n) = ------ F(n) F(n + 1) + ------- F(n) F(n + 1) 83182 83182 1407225 2 5 14496 3 1830 3 - ------- F(n) F(n + 1) - ----- F(n) - ----- F(n + 1) 83182 41591 41591 76611 3 7881 3 195735 2 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 83182 41591 83182 1215 2 2 382075 3 4 638727 2 + ---- F(n) F(n + 1) - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 7562 83182 83182 14496 4 1830 4 + ----- F(n) + ----- F(n + 1) 41591 41591 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 465 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 4 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 3 4 G(n) = -1/396 F(n) (371 F(n) - 371 F(n) - 873 F(n) F(n + 1) + 131 F(n + 1) 3 2 2 5 - 262 F(n) F(n + 1) + 1004 F(n) F(n + 1) - 16250 F(n) F(n + 1) 6 3 3 + 2099 F(n) F(n + 1) + 7150 F(n + 1) + 18850 F(n) F(n + 1) 2 4 2 - 4550 F(n) F(n + 1) - 7299 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 466 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 4 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 690 3 48825 8 9303 2 G(n) = ----- F(n + 1) + ----- F(n + 1) + ----- F(n) F(n + 1) 6061 24244 12122 3411 2 2955 3 47185 7 - ----- F(n) F(n + 1) - ---- F(n) - ----- F(n) F(n + 1) 12122 6061 12122 12323 3 176085 6 2 253 2 2 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) - --- F(n) F(n + 1) 12122 24244 116 2955 28549 3 1825 2 6 57885 4 + ---- - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n + 1) 6061 12122 12122 24244 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 467 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 4 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 113967 6 2 48747 4 42789 2 3 G(n) = ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 10945 83182 41591 202055 2 6 7829 3 19401 3 2 + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 83182 3781 7562 111183 4 850667 5 3 1830 1830 + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) - ----- + ----- F(n + 1) 41591 83182 41591 41591 54861 58521 8 237627 3 264127 7 - ----- F(n) + ----- F(n) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 83182 83182 415910 415910 39458 2 2 - ------ F(n) F(n + 1) 207955 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 468 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 49850 9 8409 3 394107 2 3 G(n) = ------ F(n + 1) + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) 319 6061 12122 837 2 2 19995 3 919635 5 + --- F(n) F(n + 1) - ----- F(n) F(n + 1) + ------ F(n + 1) 418 6061 6061 12063 4 123129 3 2 755767 4 - ----- F(n + 1) - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 12122 6061 12122 11175 8 100450 3 6 6225 2 7 + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 29 319 319 10683 10683 28205 + ----- - ----- F(n) + ----- F(n + 1) 12122 12122 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 469 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 4 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 690 9 117449 9 153147 8 2 G(n) = ----- F(n + 1) + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 6061 30305 6061 20898 5 4 5167 5 1406699 4 6 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ------- F(n) F(n + 1) 6061 60610 60610 52101 4 5 1953 10 19518 9 43929 5 + ----- F(n) F(n + 1) + ---- F(n) - ----- F(n) - ----- F(n) 6380 638 30305 60610 10257 2 690 10 962429 4 2 - ----- F(n) + ---- F(n + 1) - ------ F(n) F(n + 1) 6061 6061 60610 138843 4 52645 3 7 10611 2 7 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 60610 12122 24244 6489 2 3 2925 8 1657 9 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 2204 12122 12122 192255 6 3 408469 5 5 - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 24244 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 470 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 5 G(n) = 1/45 F(n) F(n + 1) (-9 F(n + 1) - 759 F(n + 1) F(n) 3 6 2 7 8 - 6 F(n + 1) F(n) - 273 F(n + 1) F(n) + 556 F(n + 1) F(n) 5 4 5 5 9 + 551 F(n + 1) F(n) - 25 F(n) + 44 F(n + 1) - 68 F(n) - 11 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 471 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 3 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 13242015 4 7 1012105 4 3 G(n) = -------- F(n) F(n + 1) + ------- F(n) F(n + 1) 12122 12122 98541 2 1177155 3 8 16525 3 - ----- F(n) F(n + 1) + ------- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 12122 12122 874685 2 9 1428600 2 5 21985 2 2 - ------ F(n) F(n + 1) + ------- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 6061 12122 14533 2 230805 10 6513035 6 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) - ------- F(n) F(n + 1) 12122 418 12122 3037 3 1669975 3 4 1070 1070 11 + ---- F(n) F(n + 1) - ------- F(n) F(n + 1) - ---- + ---- F(n + 1) 6061 12122 6061 6061 27 4 2113 3 + ----- F(n) + ----- F(n) 12122 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 472 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 3 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 4 6 G(n) = -1/84 F(n) (438 F(n) F(n + 1) - 79500 F(n) F(n + 1) 2 8 5 2 4 + 93375 F(n) F(n + 1) + 38328 F(n) F(n + 1) - 17346 F(n) F(n + 1) 4 2 3 3 3 7 - 2019 F(n) F(n + 1) + 6927 F(n) F(n + 1) - 1875 F(n) F(n + 1) 2 2 3 2 + 108 F(n) F(n + 1) - 108 F(n) F(n + 1) + 34 - 78 F(n + 1) 9 2 4 - 38250 F(n) F(n + 1) - 31 F(n) - 3 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 473 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 3 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2877 2 2 1070 7 159635 2 5 G(n) = ----- F(n) F(n + 1) - ---- F(n + 1) - ------ F(n) F(n + 1) 12122 6061 12122 189785 4 3 2317 4 78305 6 + ------ F(n) F(n + 1) + ---- F(n) + ----- F(n) F(n + 1) 12122 6061 12122 1070 4 2317 3 70437 2 + ---- F(n + 1) - ---- F(n) - ----- F(n) F(n + 1) 6061 6061 12122 22949 2 9959 3 48155 3 4 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 12122 12122 12122 2865 3 - ---- F(n) F(n + 1) 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 474 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 3 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 4 2 2 G(n) = 2/231 F(n) (-104 F(n) - 20 F(n + 1) - 281 F(n) F(n + 1) 5 2 3 3 + 1120 F(n) F(n + 1) + 56 F(n + 1) - 2520 F(n) F(n + 1) 3 2 4 3 + 40 F(n) F(n + 1) + 1400 F(n) F(n + 1) + 261 F(n) F(n + 1) 5 2 6 - 56 F(n) F(n + 1) - 784 F(n) + 888 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 475 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1830 7 33537 3 33537 2 24224 4 2 G(n) = ----- F(n + 1) - ----- F(n) + ----- F(n) + ----- F(n) F(n + 1) 41591 41591 41591 2189 40219 2 4 15651 5 1394559 2 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ------- F(n) F(n + 1) 4378 4378 83182 332399 3040485 2 5 415455 2 - ------ F(n) F(n + 1) - ------- F(n) F(n + 1) + ------ F(n) F(n + 1) 83182 83182 83182 1410855 6 802005 3 4 18943 3 3 + ------- F(n) F(n + 1) - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 83182 83182 4378 3649335 4 3 1830 2 + ------- F(n) F(n + 1) - ----- F(n + 1) 83182 41591 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 476 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 534650 8 2283 4 91090 2 2 G(n) = ------- F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 418 6061 425 2 3 3329 3 2 522 5 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n + 1) 209 418 209 224130 3 1060141 4 7019 16208 - ------ F(n) F(n + 1) + ------- F(n + 1) - ----- F(n) + ----- F(n + 1) 6061 12122 12122 6061 7019 1302200 7 2237625 3 5 + ----- + ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 12122 6061 12122 12500 2 6 31496 3 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 477 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 3 G(n) = -1/231 F(n) (334 F(n) - 334 F(n) - 1736 F(n + 1) - 22321 F(n + 1) 3 4 2 3 2 + 51425 F(n) F(n + 1) + 1988 F(n) F(n + 1) - 2241 F(n) F(n + 1) 6 2 7 - 52475 F(n) F(n + 1) + 8169 F(n) F(n + 1) + 22225 F(n + 1) 3 2 2 5 4 - 5138 F(n) F(n + 1) - 5450 F(n) F(n + 1) + 3794 F(n) F(n + 1) + 1760 F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 478 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 36463 5 27217 8 32919 3 2 G(n) = ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 12122 12122 60610 27052 3 3 639 1070 2 - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n + 1) 6061 6061 6061 11653 6 1759 9 25073 4 2 595 2 4 + ----- F(n) - ---- F(n) + ----- F(n) F(n + 1) + --- F(n) F(n + 1) 12122 1102 6061 638 3848 43061 7 2 344823 6 3 + ---- F(n) - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 6061 3190 12122 584933 2 7 10197 3 6 386253 2 3 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 60610 551 60610 1070 9 - ---- F(n + 1) 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 479 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 4 2 3 7 G(n) = 1/84 F(n) (-105 F(n) + 19365 F(n) F(n + 1) - 157125 F(n) F(n + 1) 3 3 2 2 3 + 25575 F(n) F(n + 1) - 16455 F(n + 1) - 11 F(n) - 129 F(n) F(n + 1) 2 8 3 - 13875 F(n) F(n + 1) - 375 F(n) F(n + 1) + 258 F(n) F(n + 1) 5 9 4 - 70830 F(n) F(n + 1) + 197250 F(n) F(n + 1) - 129 F(n + 1) 6 10 + 95970 F(n + 1) - 79500 F(n + 1) + 116) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 480 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 2 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 674 6 1282 2 608 3 612 4 6 G(n) = ---- F(n) + ---- F(n) - --- F(n) - --- F(n) F(n + 1) 319 319 319 319 81535 4 3 9717 4 7 4706 5 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 9660 5 2 316 5 5 40651 5 6 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) - ----- F(n) F(n + 1) 29 319 29 42337 6 326 2 4 167730 2 5 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) + ------ F(n) F(n + 1) 319 319 319 612 2 8 19628 2 9 356 3 7 + --- F(n) F(n + 1) - ----- F(n) F(n + 1) + --- F(n) F(n + 1) 319 29 319 525496 3 8 4784 4 2 46807 2 + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 319 379 10 356 9 20 10 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n + 1) 319 319 319 20 11 - --- F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 481 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 17525 7 2725 2 5 1088 2 G(n) = ------ F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - 9/319 638 638 319 4 185 3 17485 3 4 + 7/22 F(n) - --- F(n) + ----- F(n + 1) + 1/11 F(n + 1) 638 638 39075 3 4 14 3 1900 6 - ----- F(n) F(n + 1) - -- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 11 29 3416 2 - ---- F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 482 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 2 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 4 6 G(n) = 1/420 F(n) (329 F(n + 1) - 763 F(n) F(n + 1) - 395 F(n) + 395 F(n) 2 3 2 2 - 224 F(n + 1) + 230 F(n) F(n + 1) - 1005 F(n) F(n + 1) 3 3 3 4 2 + 890 F(n) F(n + 1) - 3290 F(n) F(n + 1) + 4375 F(n) F(n + 1) 5 4 - 427 F(n) F(n + 1) - 115 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 483 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 125795 3 4 585965 4 3 G(n) = ------- F(n) F(n + 1) + ------ F(n) F(n + 1) 12122 12122 489165 2 5 222595 6 169 5 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - --- F(n) F(n + 1) 12122 12122 551 65171 2 48349 5 18461 2 4 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 12122 12122 12122 19975 4 2 225299 2 5246 2 + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) + ---- F(n) 6061 12122 6061 1070 2 1447 3 3 1070 7 5246 3 - ---- F(n + 1) - ---- F(n) F(n + 1) + ---- F(n + 1) - ---- F(n) 6061 6061 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 484 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3175 3 5 20 391 4 63 3 2 G(n) = ----- F(n) F(n + 1) - --- + --- F(n) - -- F(n) F(n + 1) 319 319 638 22 1985 3 53025 2 6 4085 2 2 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 638 319 11400 7 22843 3 53425 4 4 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 638 638 351 20 2 3 59 4 - --- F(n) + --- F(n + 1) - 8/11 F(n) F(n + 1) + -- F(n) F(n + 1) 638 319 22 4 + 6/11 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 485 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 32 2 2 7999 3 G(n) = -46/7 F(n + 1) + -- F(n) + 46/7 F(n + 1) + ---- F(n) F(n + 1) 21 84 18925 8 3173 2 2 2 4 + ----- F(n + 1) - ---- F(n) F(n + 1) + 6 F(n) F(n + 1) 84 84 296 3 3 6625 3 5 46075 7 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 21 14 84 5 551 3 2 6 - 54/7 F(n) F(n + 1) + --- F(n) F(n + 1) + 25/4 F(n) F(n + 1) 42 268 5 18797 4 32 + --- F(n) F(n + 1) - ----- F(n + 1) - -- 21 84 21 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 486 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 28067 3 27583 4 22717 7 16987 7 G(n) = ------ F(n + 1) + ----- F(n + 1) + ----- F(n + 1) - ----- F(n) 30305 24244 30305 12122 7005 4 20696 3 52621 6 + ---- F(n) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 2552 6061 12122 478109 5 3 473907 5 2 122345 4 4 - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 24244 60610 12122 19742 4 3 143795 7 948625 6 2 + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 6061 24244 48488 18541 8 65147 16571 6 30936 2 + ----- F(n + 1) - ----- - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 48488 48488 6061 30305 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 487 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2251 2 3523 52 2 4 G(n) = ----- F(n + 1) + ---- F(n) F(n + 1) + -- F(n) F(n + 1) 638 1595 11 156843 5 4 264622 6 3 94773 7 2 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 7975 7975 15950 9978 8 79 6 211 6 4779 2 + ---- F(n) F(n + 1) + -- F(n + 1) + --- F(n) - ---- F(n) 7975 22 55 1595 39327 9 613 9 98541 4 527 5 - ----- F(n) + ---- F(n + 1) - ----- F(n) F(n + 1) - --- F(n) F(n + 1) 15950 1450 15950 55 2161 5 2357 15757 + ---- F(n + 1) + ---- F(n) - ----- F(n + 1) 1450 1450 7975 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 488 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 25211 5 10417 5 1721 5 2 G(n) = ----- F(n) - ----- F(n + 1) - ---- F(n) F(n + 1) 378 1512 84 30089 6 3 7097 7 2 24979 8 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 378 84 168 230 2 3 41 2 5 3155 6 + --- F(n) F(n + 1) + -- F(n) F(n + 1) + ---- F(n) F(n + 1) 189 42 168 719 223 3 41 3 6937 9 673 9 + --- F(n + 1) - --- F(n) + --- F(n + 1) - ---- F(n) + --- F(n + 1) 216 42 168 108 189 239 7 41 7 41 2 8377 2 7 + --- F(n) - --- F(n + 1) + -- F(n) F(n + 1) - ---- F(n) F(n + 1) 84 168 56 504 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 489 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 887 3 887 2 17950 10 10053 2 G(n) = ---- F(n) + --- F(n) - ----- F(n + 1) - ----- F(n) F(n + 1) 638 638 11 319 4659 4075 6 396069 5 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 319 22 638 2979 2 44675 9 131761 2 4 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 319 11 319 3755 3 4 102301 3 3 35100 3 7 - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 22 319 11 7175 2 8 70 2 5 494708 6 - ---- F(n) F(n + 1) + -- F(n) F(n + 1) + ------ F(n + 1) 22 11 319 25822 2 49195 3 1695 7 + ----- F(n + 1) + ----- F(n + 1) - ---- F(n + 1) 319 638 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 490 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 5 G(n) = 1/12 F(n) (32 F(n) F(n + 1) - 165 F(n) F(n + 1) - 47 F(n) F(n + 1) 2 4 2 4 6 4 + 175 F(n) F(n + 1) - 41 F(n) - 15 F(n) + 56 F(n) + F(n + 1) 6 3 2 2 3 + 5 F(n + 1) - 2 F(n) F(n + 1) - 25 F(n) F(n + 1) + 26 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 491 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 56875 6 2837 2 39 2 4 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + -- F(n) F(n + 1) 638 638 22 4025 2 5 26425 3 4 2425 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 319 638 3576 2 301 2 11900 7 301 3 + ---- F(n + 1) + --- F(n) - ----- F(n + 1) - --- F(n) 319 638 319 638 11920 3 124 6 577 5 + ----- F(n + 1) - --- F(n + 1) + --- F(n) F(n + 1) 319 11 22 589 3 3 9245 2 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 22 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 492 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 60050 8 666885 2 372225 6 G(n) = ------ F(n + 1) - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 319 12122 1102 175350 3 4 78225 7 12800 2 6 - ------ F(n) F(n + 1) - ----- F(n + 1) - ----- F(n) F(n + 1) 551 551 319 232217 3 2251153 4 32127 3 - ------ F(n) F(n + 1) + ------- F(n + 1) - ----- F(n) 12122 12122 12122 859785 3 32127 528502 3 149950 7 + ------ F(n + 1) + ----- - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 6061 12122 6061 319 263837 2 2 21525 2 5 116300 3 5 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 6061 1102 319 194709 2 + ------ F(n) F(n + 1) 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 493 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 4 G(n) = -1/12 F(n) (16 F(n) - 16 F(n) - 88 F(n + 1) + 192 F(n) F(n + 1) 3 2 2 5 3 4 - 264 F(n) F(n + 1) - 300 F(n) F(n + 1) + 2625 F(n) F(n + 1) 2 3 2 6 + 104 F(n) F(n + 1) - 110 F(n) F(n + 1) - 2650 F(n) F(n + 1) 2 7 3 + 409 F(n) F(n + 1) + 1125 F(n + 1) - 1131 F(n + 1) + 88 F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 494 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 133 3 9357 3 5 103 2 5 G(n) = ---- F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 1276 1276 116 877 3 4 90419 4 3247 4 21559 8 3073 - --- F(n) F(n + 1) + ----- F(n) - ---- F(n + 1) - ----- F(n) + ---- 319 5104 2552 1276 5104 2202 5 2 129 8 5341 4 3 - ---- F(n) F(n + 1) + --- F(n + 1) + ---- F(n) F(n + 1) 319 176 1276 60691 5 3 1371 6 156105 6 2 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) 2552 319 5104 7 213 7 907 7 - 355/8 F(n) F(n + 1) - ---- F(n + 1) - --- F(n) 1276 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 495 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 17267 131 109 3 3937 9 7 G(n) = ------ F(n) - ---- F(n + 1) + --- F(n) + ---- F(n) - 35/3 F(n) 2460 2460 12 410 7 17167 4 2 5 + 10 F(n + 1) + ----- F(n) F(n + 1) + 30 F(n) F(n + 1) 2460 119 2 36901 7 2 6579 8 - --- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 12 820 410 179 2 6 5429 8 + --- F(n) F(n + 1) - 35 F(n) F(n + 1) + ---- F(n) F(n + 1) 12 164 248779 6 3 3351 5 3 40081 9 - ------ F(n) F(n + 1) + ---- F(n + 1) - 10 F(n + 1) - ----- F(n + 1) 2460 205 2460 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 496 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 604 27921 2 2 135535 8 G(n) = ---- F(n) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 319 638 319 2425 7 5325 4 4 70755 3 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 22 22 638 2925 2 6 7895 3 6 41137 3 2 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 11 638 638 4207 3 326995 2 7 582215 4 5 - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 319 319 638 20 9 624 4 150 3 5 6375 4 + --- F(n + 1) + --- F(n) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 11 319 111433 2 3 134819 4 20 + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) - --- 638 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 497 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 n - j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 7 G(n) = -1/24 F(n) (-99 F(n) F(n + 1) + 40 F(n) - 559 F(n) + 465 F(n) 7 9 4 4 3 + 4 F(n + 1) + 54 F(n) - 466 F(n) F(n + 1) - 130 F(n) F(n + 1) 7 2 8 6 + 1410 F(n) F(n + 1) - 232 F(n) F(n + 1) + 85 F(n) F(n + 1) 2 7 5 3 2 - 228 F(n) F(n + 1) - 16 F(n + 1) + 918 F(n) F(n + 1) 5 2 5 4 6 - 1192 F(n) F(n + 1) + 2934 F(n) F(n + 1) + 1520 F(n) F(n + 1) 6 3 - 4508 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 498 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 20 3 273 2 256 3 G(n) = - --- + --- F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 319 333 2 236 2 2 202 3 + --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) 319 319 319 100 3 222 4 - --- F(n) F(n + 1) + --- F(n) 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 499 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = -1/42 F(n) (-19 - 3 F(n + 1) + 3 F(n) F(n + 1) - 2 F(n) 2 2 3 4 - 81 F(n) F(n + 1) + 81 F(n) F(n + 1) + 21 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 500 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5571 2 7641 2 4937 5 G(n) = ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 6061 6061 2748 2 4 41183 3 3 4404 3 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + ---- F(n) 551 6061 6061 690 3 17962 5 5431 6 491 2 - ---- F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) - --- F(n) 6061 6061 12122 418 690 2 + ---- F(n + 1) 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 501 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 20 689 643 3 G(n) = --- - --- F(n + 1) + --- F(n) - --- F(n) F(n + 1) 319 319 638 638 1543 2 2 164 2 3 926 3 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 319 658 3 2 695 4 326 4 712 5 + --- F(n) F(n + 1) + --- F(n) - --- F(n) F(n + 1) - --- F(n) 319 638 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 502 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 53 2 2 G(n) = 44/7 F(n + 1) + -- F(n) F(n + 1) - 1/21 F(n) F(n + 1) 42 69 2 4 3 3 - -- F(n) F(n + 1) - 1/14 F(n) F(n + 1) + 2/21 F(n) F(n + 1) + 1/42 14 4 187 5 239 3 3 2 - 1/42 F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) - 1/42 F(n) 14 14 2 - 44/7 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 503 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 20 2 1845 2 1139 5 1492 6 G(n) = --- F(n + 1) - --- F(n + 1) - ---- F(n) - ---- F(n) + ---- F(n) 319 319 638 638 319 2597 2 3 4603 3 2 533 3 3 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 638 638 319 410 4 433 5 829 5 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 1732 4 1048 4 2 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 504 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = -1/84 F(n) (-59 + 57 F(n + 1) - 57 F(n) F(n + 1) + 38 F(n) 2 2 3 4 - 141 F(n) F(n + 1) + 141 F(n) F(n + 1) + 21 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 505 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5729 5 453 5 4734 2 G(n) = ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 6061 6061 1524 2 2086 2 1070 2 1070 3 + ---- F(n) F(n + 1) - ---- F(n) + ---- F(n + 1) - ---- F(n + 1) 6061 6061 6061 6061 2086 3 21205 2 4 21008 3 3 + ---- F(n) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 12122 6061 16679 4 2 + ----- F(n) F(n + 1) 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 506 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 20 631 585 3 77 2 2 G(n) = --- - --- F(n + 1) + --- F(n) - --- F(n) F(n + 1) + -- F(n) F(n + 1) 319 319 638 638 58 280 2 3 491 3 252 3 2 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 173 4 152 4 422 5 + --- F(n) - --- F(n) F(n + 1) - --- F(n) 638 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 507 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 4 3 G(n) = 1/84 F(n) (-351 F(n) F(n + 1) + 463 F(n) F(n + 1) - 66 F(n + 1) 5 2 2 + 68 F(n + 1) + 99 F(n) F(n + 1) - 107 F(n) F(n + 1) 2 3 3 5 - 106 F(n) F(n + 1) - 170 F(n) + 37 F(n) + 133 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 508 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 19365 2 2105 4 3 7815 2 5 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 12122 1102 1102 1945 6 8275 3 4 19307 3 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 1102 1102 12122 2086 3 18313 2 21231 2 2 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 12122 12122 4483 3 1070 3 1070 4 4483 4 + ---- F(n) + ---- F(n + 1) - ---- F(n + 1) - ---- F(n) 6061 6061 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 509 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 1227 189 4 8094 5 G(n) = ----- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 22 319 93 2 3 62 2 4 144 3 2 + -- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 22 319 11 6718 3 3 48 5 3246 6 617 - ---- F(n) F(n + 1) - -- F(n + 1) - ---- F(n + 1) - --- F(n) 319 11 319 638 617 2 1412 3226 2 + --- F(n) + ---- F(n + 1) + ---- F(n + 1) 638 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 510 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 6 5 G(n) = -1/84 F(n) (3165 F(n) F(n + 1) + 1110 F(n + 1) - 2475 F(n) F(n + 1) 4 3 + 75 F(n + 1) + 332 F(n) F(n + 1) - 150 F(n) F(n + 1) 2 2 2 4 3 + 275 F(n) F(n + 1) - 945 F(n) F(n + 1) - 200 F(n) F(n + 1) 4 2 2 + 75 F(n) - 75 F(n) - 1187 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 511 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 949 3 415 3 903 2 415 2 G(n) = ----- F(n + 1) - --- F(n) + ---- F(n + 1) + --- F(n) 1276 319 1276 319 823 6 2375 2 4 79 7 - ---- F(n + 1) + ---- F(n) F(n + 1) + --- F(n + 1) 1276 1276 116 437 2 5 60 3 3 313 4 3 - --- F(n) F(n + 1) - -- F(n) F(n + 1) - --- F(n) F(n + 1) 638 11 58 5173 5 3273 2 41 3 4 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + -- F(n) F(n + 1) 638 1276 22 10675 4 2 12911 6 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 1276 1276 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 512 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 2 2 G(n) = 2/15 F(n + 1) (-72 F(n + 1) - 22 F(n + 1) F(n) - 8 F(n) 6 3 3 + 27 F(n + 1) F(n) + 72 F(n + 1) + 170 F(n + 1) F(n) 5 - 167 F(n + 1) F(n)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 513 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 8982 2 2061 3 7818 G(n) = ------ F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- 41591 3781 41591 369625 2 6 138425 8 1503379 4 + ------ F(n) F(n + 1) - ------ F(n + 1) + ------- F(n + 1) 83182 7562 83182 7818 3 1830 3 3589625 7 - ----- F(n) + ----- F(n + 1) + ------- F(n) F(n + 1) 41591 41591 83182 1763075 3 5 483625 3 14472 2 - ------- F(n) F(n + 1) - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 41591 83182 41591 100523 2 2 + ------ F(n) F(n + 1) 83182 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 514 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 23 6 6 4 25 4 G(n) = --- F(n + 1) - 5/14 F(n) - 5/84 F(n + 1) + -- F(n) 56 84 3 3 41 2 2 + 24/7 F(n) F(n + 1) + -- F(n) F(n + 1) + 5/21 F(n) F(n + 1) 28 25 3 23 2 331 4 2 - -- F(n) F(n + 1) + 5/84 + -- F(n + 1) - --- F(n) F(n + 1) 28 56 56 97 2 4 + -- F(n) F(n + 1) 56 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 515 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1070 3 658 4 10778 2 5 G(n) = ---- F(n + 1) - ---- F(n) + ----- F(n) F(n + 1) 6061 6061 6061 3666 4 3 16975 3 1337 6 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 551 12122 1102 25299 5 2 1070 32863 3 4 - ----- F(n) F(n + 1) - ---- - ----- F(n) F(n + 1) 12122 6061 6061 1223 2 2 497 3 54 2 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 1102 6061 209 1728 7 + ---- F(n) 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 516 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 43 4 4 92 6 2 G(n) = 1/25 F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 15 15 7 184 3 776 5 3 8 - 14/5 F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) - F(n) 75 75 26 4 + 1 - -- F(n + 1) 25 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 517 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 295 2 1055 3 13 3 2425 3 4 G(n) = --- F(n) F(n + 1) - ---- F(n + 1) + -- F(n) + ---- F(n) F(n + 1) 21 28 42 28 6 1055 7 325 2 - 620/7 F(n) F(n + 1) + ---- F(n + 1) - --- F(n) F(n + 1) 28 84 4 25 2 3 115 2 5 + 25/6 F(n) F(n + 1) + -- F(n) F(n + 1) - --- F(n) F(n + 1) 12 14 3 2 13 25 25 5 - 25/4 F(n) F(n + 1) - -- F(n) + -- F(n + 1) - -- F(n + 1) 42 12 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 518 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 170261 7 3814 40159 5813 8 71571 G(n) = ------- F(n) F(n + 1) + ---- F(n) - ----- F(n + 1) + ---- F(n) - ----- 5510 6061 12122 1102 12122 11187 4 8944 8 101464 6 2 - ----- F(n + 1) + ---- F(n + 1) + ------ F(n) F(n + 1) 1102 551 2755 27895 7 2 3 207 3 2 + ----- F(n) F(n + 1) - 7/2 F(n) F(n + 1) + --- F(n) F(n + 1) 1102 22 73 4 62274 2 2 76119 3 - -- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 11 2755 5510 69 5 + -- F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 519 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 3 2 G(n) = -1/84 F(n) (-280 F(n + 1) - 1015 F(n) F(n + 1) + 322 F(n + 1) 7 2 6 + 6225 F(n + 1) + 2229 F(n) F(n + 1) - 14550 F(n) F(n + 1) 2 2 3 3 4 - 530 F(n) F(n + 1) + 490 F(n) F(n + 1) + 14925 F(n) F(n + 1) 3 2 5 4 - 6311 F(n + 1) - 2100 F(n) F(n + 1) + 595 F(n) F(n + 1) - 56 F(n) 3 + 56 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 520 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 8 4 G(n) = -1/396 F(n) (177 F(n + 1) + 12125 F(n + 1) - 12166 F(n + 1) 3 2 2 - 177 F(n) F(n + 1) + 4607 F(n) F(n + 1) - 1350 F(n) F(n + 1) 3 7 2 6 + 259 F(n) F(n + 1) - 28775 F(n) F(n + 1) - 2350 F(n) F(n + 1) 3 5 2 + 27650 F(n) F(n + 1) - 118 + 118 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 521 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3969 3 2497 5 2 4739 2 G(n) = ----- F(n) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 12122 1102 6061 2589 4 2067 3 13455 2 2 + ----- F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 12122 12122 6061 4960 2 5 10497 3 40125 2 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 551 6061 12122 5110 6 2889 7 30399 3 3969 - ---- F(n) F(n + 1) + ---- F(n + 1) - ----- F(n + 1) - ----- 551 1102 12122 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 522 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1196495 7 2 164679 8 G(n) = -------- F(n) F(n + 1) - ------ F(n) F(n + 1) 83182 83182 527363 4 5 251489 3 2 881 + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - ---- F(n) 41591 83182 2189 29037 8 42951 4 1830 1830 9 32906 9 - ------ F(n) - ------ F(n) + ----- - ----- F(n + 1) + ----- F(n) 166364 166364 41591 41591 41591 3084 4 4227 3 12405 7 + ---- F(n) F(n + 1) - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 3781 166364 166364 120375 6 2 62025 5 3 145185 4 4 + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 166364 166364 166364 203041 4 423627 3 6 6321 3 5 + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 41591 83182 8756 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 523 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 690 3054 2374 2 6 1008 4 G(n) = ----- F(n + 1) + ---- F(n) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 6061 551 319 4601 4 4 220 7 372 5 3 690 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- 551 551 19 6061 1287 8 351 5 2367 3 2 13786 6 2 - ---- F(n) + --- F(n) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 1102 638 638 551 6186 7 1053 2 3 6068 3 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 551 638 551 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 524 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(4 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 8373 2 8373 5 70125 2 7 1011 G(n) = ----- F(n) + ---- F(n) - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 6061 6061 1102 1102 690 5 690 6 3840 5 + ---- F(n + 1) - ---- F(n + 1) + ---- F(n) F(n + 1) 6061 6061 551 7155 4 2 3407 4 33525 3 6 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 638 551 210994 4 37775 8 53804 3 2 - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 1102 6061 10793 2 3 27345 3 3 1695 5 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 12122 6061 6061 66925 4 5 + ----- F(n) F(n + 1) 551 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 525 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(n + j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1475 3 2 972 3 3 878 2 3 G(n) = ----- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 912 2 4 381 4 404 5 - --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 862 4 850 4 2 1198 5 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 20 20 2 236 2 236 5 + --- F(n + 1) - --- F(n + 1) + --- F(n) - --- F(n) 319 319 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 526 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(n + j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 4 2 4 G(n) = 1/84 F(n) (-1610 F(n) F(n + 1) - 12 F(n + 1) + 945 F(n) F(n + 1) 6 4 6 2 5 - 28 F(n + 1) - 61 F(n) + 460 F(n) - 399 F(n) + 609 F(n) F(n + 1) 2 2 2 3 - 149 F(n) F(n + 1) + 84 F(n + 1) + 24 F(n) F(n + 1) 3 + 137 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 527 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 685 7 685 6 131 6 49 2 2179 3 G(n) = ---- F(n) + --- F(n) + --- F(n + 1) - -- F(n + 1) + ---- F(n + 1) 638 638 29 11 638 475 3 4 144 5 2 6703 5 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 29 638 6 2219 7 4 2 + 17/2 F(n) F(n + 1) - ---- F(n + 1) + 14 F(n) F(n + 1) 638 1255 4 3 3397 5 3101 6 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 638 638 364 3 3 - --- F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 528 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 5 3 3 G(n) = -1/84 F(n) (1110 F(n + 1) - 2475 F(n) F(n + 1) + 3165 F(n) F(n + 1) 2 2 4 4 - 1187 F(n + 1) - 19 F(n) + 19 F(n) + 75 F(n + 1) + 332 F(n) F(n + 1) 3 2 2 2 4 - 150 F(n) F(n + 1) + 191 F(n) F(n + 1) - 945 F(n) F(n + 1) 3 - 116 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 529 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(n + 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3199 1059 5 83 3 2 G(n) = ------ F(n + 1) + ----- F(n + 1) + -- F(n) F(n + 1) 12122 12122 22 35779 3 637785 2 6 88599 2 2 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 24244 12122 259575 7 9281 5 567977 3 5 + ------ F(n) F(n + 1) + ----- F(n) - ------ F(n) F(n + 1) 12122 12122 12122 25561 4 73090 8 9967 8 28529 - ----- F(n) F(n + 1) - ----- F(n + 1) + ----- F(n) - ----- 12122 6061 24244 24244 325169 4 13943 2 3 + ------ F(n + 1) - ----- F(n) F(n + 1) 24244 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 530 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(n + 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 6422 2 20 2 13 2 20 3 G(n) = ----- F(n) + --- F(n + 1) - -- F(n) F(n + 1) - --- F(n + 1) 1595 319 29 319 651 6 1213 6 129 2 4 + --- F(n) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 110 638 11 733 2 5 301 3 3 160 3 4 - --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 22 319 1633 6 919 2 1207 7 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) 319 638 638 196 5 633 - --- F(n) F(n + 1) + --- F(n) F(n + 1) 55 290 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 531 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(n + 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 4 6 G(n) = -1/84 F(n) (-86 F(n + 1) - 42 F(n) F(n + 1) + 129 F(n) F(n + 1) 2 3 2 + 476 F(n) F(n + 1) + 84 F(n + 1) - 681 F(n) F(n + 1) 2 5 4 3 6 - 182 F(n) F(n + 1) - 2030 F(n) F(n + 1) + 263 F(n) F(n + 1) 5 2 3 2 4 + 2475 F(n) F(n + 1) - 546 F(n) F(n + 1) + 140 F(n) F(n + 1) 5 7 - 56 F(n) + 56 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 532 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 7705 6 3610 3 4 1679 G(n) = ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- 22 11 638 12800 2 6 12457 3 118381 4 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ------ F(n + 1) 319 638 638 1679 3 93855 3 60050 8 3235 7 - ---- F(n) + ----- F(n + 1) - ----- F(n + 1) - ---- F(n + 1) 638 638 319 22 36815 2 27648 3 149950 7 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 638 319 319 10809 2 13819 2 2 415 2 5 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + --- F(n) F(n + 1) 638 319 22 116300 3 5 - ------ F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 533 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 407 4 11 9 59 9 8 G(n) = --- F(n) F(n + 1) - -- F(n + 1) + -- F(n) - 2/75 F(n) F(n + 1) 75 75 75 47 7 2 1873 6 3 44 5 59 + -- F(n) F(n + 1) - ---- F(n) F(n + 1) - -- F(n + 1) - -- F(n) 25 75 25 75 143 1327 5 4 + --- F(n + 1) + ---- F(n) F(n + 1) 75 75 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 534 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(n + 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 221 3 439 3 3 12025 7 G(n) = --- F(n) F(n + 1) - --- F(n) F(n + 1) - ----- F(n) F(n + 1) 28 12 28 757 2 2 2 4 5325 3 5 - --- F(n) F(n + 1) + 23/4 F(n) F(n + 1) + ---- F(n) F(n + 1) 28 14 2477 4 125 2 6 407 5 - ---- F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 14 28 12 55 2057 3 4975 8 - -- F(n) F(n + 1) + ---- F(n) F(n + 1) - 3/4 + ---- F(n + 1) 12 28 28 6 2 2 - 89/6 F(n + 1) + 3/4 F(n) + 89/6 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 535 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(n + 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 18359239 119621 267 6 1511452 5 G(n) = -------- F(n) - ------ F(n + 1) + --- F(n + 1) - ------- F(n + 1) 2485010 65395 11 112955 1231 5 651 3 3 111 2 4 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 22 11 11 1380562 4 94157 6719013 8 - ------- F(n) F(n + 1) + ----- F(n) F(n + 1) - ------- F(n) F(n + 1) 112955 12122 225910 1145968 7 2 9187554 6 3 - ------- F(n) F(n + 1) + ------- F(n) F(n + 1) 112955 112955 1361443 8 1738011 9 18247 2 664482 9 - ------- F(n) F(n + 1) + ------- F(n + 1) - ----- F(n) - ------ F(n) 45182 112955 12122 112955 148187 2 - ------ F(n + 1) 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 536 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(n + 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 8 G(n) = 1/84 F(n) (-577 F(n) F(n + 1) + 709 F(n) F(n + 1) + 88 F(n + 1) 4 4 3 5 2 6 + 30463 F(n) F(n + 1) - 5674 F(n) F(n + 1) - 27688 F(n) F(n + 1) 2 4 2 2 7 + 989 F(n) F(n + 1) + 4076 F(n) F(n + 1) + 12049 F(n) F(n + 1) 3 6 3 4 - 913 F(n) F(n + 1) - 44 F(n + 1) - 12401 F(n) F(n + 1) + 124 F(n) 2 3 3 4 2 - 124 F(n) + 1347 F(n) F(n + 1) - 2424 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 537 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(2 n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4506 3 12800 2 6 3 4 G(n) = ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - 325 F(n) F(n + 1) 319 319 116300 3 5 12878 2 2 2070 4 - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n + 1) 319 319 11 912 4 18074 2 912 3 46275 3 + --- F(n) - ----- F(n) F(n + 1) - --- F(n) + ----- F(n + 1) 319 319 319 319 7 149950 7 490 2 - 145 F(n + 1) + ------ F(n) F(n + 1) + --- F(n) F(n + 1) 319 29 29298 3 60050 8 2 5 - ----- F(n) F(n + 1) - ----- F(n + 1) + 20 F(n) F(n + 1) 319 319 6 + 345 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 538 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1151 3 6 2645 4 551 4 3 G(n) = ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 56 168 42 4427 4 5 461 6 18995 6 3 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) - ----- F(n) F(n + 1) 224 84 336 3475 7 2 2113 8 1054 5 2 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 56 224 105 737 6 1567 3 131 7 131 9 - --- F(n) F(n + 1) - ---- F(n + 1) - --- F(n) + --- F(n) 84 420 84 84 475 9 1567 7 285 95 5 - --- F(n + 1) + ---- F(n + 1) - --- F(n + 1) + -- F(n + 1) 672 420 224 48 43 2 + --- F(n) F(n + 1) 210 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 539 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(2 n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 111325 8 19189 8 4933 143639 4 G(n) = ------ F(n) - ----- F(n + 1) + ---- F(n + 1) + ------ F(n + 1) 11484 2871 1914 11484 5053 5 2509 9 66163 55763 5 4 - ---- F(n + 1) - ---- F(n) - ----- + ----- F(n) F(n + 1) 1914 638 11484 1914 178 6 3 5449 7 33893 7 2 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 33 2871 957 29351 8 113591 2 6 8583 2 7 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 1914 3828 638 162445 6 2 13535 8 590021 5 3 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 2871 1914 5742 103 2 3 2597 2 2 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 5742 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 540 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(3 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 49850 9 10 3 10650 2 3 G(n) = ------ F(n + 1) + --- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 319 278 2 2 100450 3 6 11175 8 - --- F(n) F(n + 1) - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 29 240 3 6225 2 7 421 1481 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n) + ---- F(n + 1) 319 319 638 319 96553 5 217 19963 4 12437 3 2 + ----- F(n + 1) - --- - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 638 319 638 201 4 + --- F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 541 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 615 3 1170 7 4077 2 G(n) = ----- F(n + 1) - ---- F(n) - ----- F(n) F(n + 1) 12122 6061 12122 3291 5 2808 6 2645 5 5 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 24244 6061 209 3547 5 156681 8 2 25361 9 - ---- F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 1102 12122 24244 11697 53939 4 6 2340 4 3 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 24244 12122 6061 23806 4 2 30981 3 7 765 2 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 24244 12122 234 6 615 10 1170 10 - ---- F(n) F(n + 1) - ----- F(n + 1) + ---- F(n) 6061 12122 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 542 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 2 3 G(n) = -1/420 F(n) (-120 F(n + 1) + 120 F(n) F(n + 1) + 33 F(n) - 2660 F(n) 9 7 2 7 2 3 + 3035 F(n) + 12 F(n) + 6840 F(n) F(n + 1) - 5908 F(n) F(n + 1) 4 6 3 8 + 5586 F(n) F(n + 1) - 20612 F(n) F(n + 1) + 9760 F(n) F(n + 1) 7 2 8 6 + 6895 F(n) F(n + 1) - 3421 F(n) F(n + 1) + 300 F(n) F(n + 1) 2 9 3 4 + 294 F(n) F(n + 1) + 110 F(n + 1) - 240 F(n) F(n + 1) 2 5 - 24 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 543 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 185 9 1075 17673 2 1883 4 G(n) = --- F(n + 1) - ---- F(n) - ----- F(n) + ---- F(n) F(n + 1) 638 1914 7975 957 687 8 1261 2 3 8069 6 3 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 1914 710 7 2 650 9 49421 10 185 10 - --- F(n) F(n + 1) + --- F(n) + ----- F(n) - --- F(n + 1) 957 957 15950 638 20486 15984 5 3123 9 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 7975 7975 7975 121723 2 4 6265 2 7 193349 6 4 + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) 15950 1914 7975 297806 7 3 3805 8 42629 8 2 - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 7975 1914 15950 21506 9 + ----- F(n) F(n + 1) 7975 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 544 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 2 G(n) = ) F(j) F(3 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 3 G(n) = 4/3 F(n) F(n + 1) - 1/6 F(n) F(n + 1) - 4/3 F(n) F(n + 1) 4 2 5 5 - 5 F(n) F(n + 1) + 23/6 F(n) F(n + 1) + 1/3 + 8/3 F(n) F(n + 1) 6 19 6 19 2 4 4 - 2/3 F(n) - -- F(n + 1) + -- F(n + 1) - 1/3 F(n + 1) - 2/3 F(n) 12 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 545 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 233 4 183 5 249 2 3 G(n) = --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 319 31 2 4 276 3 2 62 3 3 + -- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 29 319 319 70 4 1733 4 2 73 5 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) - -- F(n) F(n + 1) 319 638 29 185 5 185 6 501 5 137 2 - --- F(n + 1) + --- F(n + 1) + --- F(n) + --- F(n) 638 638 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 546 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2076 2 882 5 2118 G(n) = ------ F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 30305 551 6061 77547 6 87317 2 12915 4 2 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 60610 30305 1102 43385 4 3 7560 5 102 5 2 - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 6061 551 11 5823 6 18039 7 2373 6 22215 2 - ----- F(n) F(n + 1) + ----- F(n) - ---- F(n) + ----- F(n) 30305 12122 1102 6061 615 2 615 3 + ----- F(n + 1) - ----- F(n + 1) 12122 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 547 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 6 G(n) = 1/24 F(n) (44 F(n) F(n + 1) - 406 F(n) F(n + 1) + 145 F(n + 1) 6 2 4 2 + 98 F(n) F(n + 1) + 113 F(n) - 99 F(n) + 10 F(n) - 127 F(n + 1) 2 2 2 4 3 - 58 F(n) F(n + 1) + 290 F(n) F(n + 1) + 12 F(n) F(n + 1) 4 - 22 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 548 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 11585 2 3 3 G(n) = -37/2 F(n) F(n + 1) - ----- F(n) F(n + 1) + 16 F(n) F(n + 1) 638 430 2 2485 2 208 3 2675 3 - --- F(n) - ---- F(n + 1) - --- F(n) + ---- F(n + 1) 319 319 319 58 69945 6 117 2 1110 + ----- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 22 319 32315 3 4 2 4 4485 2 5 - ----- F(n) F(n + 1) - 3/2 F(n) F(n + 1) + ---- F(n) F(n + 1) 319 638 6 14620 7 + 15/2 F(n + 1) - ----- F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 549 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 3 4 G(n) = -1/24 F(n) (-168 F(n) F(n + 1) - 156 F(n) F(n + 1) 4 3 5 2 4 - 568 F(n) F(n + 1) + 876 F(n) F(n + 1) - 19 F(n) F(n + 1) 5 2 2 3 + 35 F(n + 1) - 7 F(n + 1) - 232 F(n) F(n + 1) + 171 F(n) F(n + 1) 3 5 6 3 + 30 F(n) + 18 F(n) + 36 F(n) F(n + 1) - 24 F(n + 1) 2 5 + 8 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 550 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 28279 3 7 4581 2 G(n) = ------ F(n) F(n + 1) + 505 F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 11063 2 2 2 5 3 5 553 + ----- F(n) F(n + 1) + 1/2 F(n) F(n + 1) - 435 F(n) F(n + 1) + --- 319 638 7 8 132017 4 2 6 + 34 F(n + 1) - 415/2 F(n + 1) + ------ F(n + 1) - 5/2 F(n) F(n + 1) 638 3400 3 1361 3 21877 3 - ---- F(n) F(n + 1) + ---- F(n) - ----- F(n + 1) 319 638 638 6 3 4 5000 2 - 167/2 F(n) F(n + 1) + 139/2 F(n) F(n + 1) + ---- F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 551 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 45 7 3 G(n) = 99/4 F(n) F(n + 1) - -- F(n + 1) + 27/2 F(n) - 37/3 F(n) 32 25 9 9 109 4 3 17 2 5 - -- F(n + 1) + 11/6 F(n) - --- F(n) F(n + 1) + -- F(n) F(n + 1) 32 12 12 61 8 7 2 2665 6 3 + -- F(n) F(n + 1) + 525/8 F(n) F(n + 1) - ---- F(n) F(n + 1) 32 48 5 2 4 491 3 6 - 56/3 F(n) F(n + 1) - 55/4 F(n) F(n + 1) + --- F(n) F(n + 1) 24 17 2 599 4 5 35 5 - -- F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n + 1) 12 32 16 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 552 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(3 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 900 2 3 17 2 4 8425 9 2 G(n) = --- F(n) F(n + 1) - -- F(n) F(n + 1) - ---- F(n + 1) - 7/22 F(n) 11 22 22 84 3 3 16975 3 6 94 5 + -- F(n) F(n + 1) - ----- F(n) F(n + 1) - -- F(n) F(n + 1) 11 22 11 4131 5 75 6 525 2 7 + ---- F(n + 1) + -- F(n + 1) - --- F(n) F(n + 1) 11 22 11 474 3 2 3579 4 20775 8 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 11 22 22 18 84 15 40 2 + -- F(n) F(n + 1) + -- F(n + 1) - -- F(n) - -- F(n + 1) 11 11 22 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 553 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(3 n - j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 3 6 G(n) = -1/28 F(n) (6 F(n + 1) - 6019 F(n + 1) + 12575 F(n) F(n + 1) 3 3 2 2 7 - 18 F(n) + 18 F(n) + 643 F(n) F(n + 1) + 550 F(n) F(n + 1) 2 3 8 4 - 1223 F(n) F(n + 1) - 15100 F(n) F(n + 1) + 2521 F(n) F(n + 1) 2 9 2 - 9 F(n) F(n + 1) + 6150 F(n + 1) + 39 F(n) F(n + 1) - 133 F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 554 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(3 n - j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 32 4 105 151 6 273 5 20 10 G(n) = ---- F(n) F(n + 1) - --- F(n) + --- F(n) + --- F(n) - --- F(n + 1) 319 319 319 319 319 20 9 41 119 5 + --- F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 823 3 6 71 3 7 78 4 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 319 6686 4 2 1246 4 5 4417 4 6 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 1578 5 863 5 4 849 5 5 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 29 1146 6 3 822 6 4 623 7 2 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 29 319 19029 7 3 - ----- F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 555 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(3 n - j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 999 6 4 10 2 5 24 5 2 G(n) = --- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 11 178 4 6 4 3 507 8 2 + --- F(n) F(n + 1) + 5/22 F(n) F(n + 1) + --- F(n) F(n + 1) 11 22 183 9 109 3 7 943 7 3 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 22 22 22 20 6 333 5 5 6 - -- F(n) F(n + 1) - --- F(n) F(n + 1) - 2/11 F(n) F(n + 1) 11 11 13 9 18 2 8 17 3 4 19 10 - -- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) 22 11 22 11 10 7 3 + 5/22 F(n + 1) + 3/11 F(n) - 5/22 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 556 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(3 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 15 6 6 5 5 G(n) = -- F(n) + 5/22 F(n + 1) - 5/22 F(n + 1) + 7/22 F(n) 22 10 2 3 24 2 4 69 5 - -- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 11 11 22 4 5 3 2 + 3/11 F(n) F(n + 1) - 9/22 F(n) F(n + 1) - 3/11 F(n) F(n + 1) 61 3 3 13 4 51 4 2 - -- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 22 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 557 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(3 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 564 6 31 7 74 7 2199 3 4 G(n) = --- F(n) + -- F(n + 1) + --- F(n) + ---- F(n) F(n + 1) 319 58 319 319 2886 4 2 1854 5 2 1194 2 4 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 319 161 2 5 3128 3 3 3687 4 3 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 58 319 638 1956 5 958 6 301 3 162 2 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n + 1) + --- F(n + 1) 319 319 638 319 182 6 - --- F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 558 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(3 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 495 5 3 3 G(n) = 2095/4 F(n + 1) - 27/7 + --- F(n) F(n + 1) - 270/7 F(n) F(n + 1) 14 477 3 2 6 435 6 + --- F(n) F(n + 1) + 135/4 F(n) F(n + 1) - --- F(n + 1) 14 28 2 4 2661 2 2 7 + 45/7 F(n) F(n + 1) - ---- F(n) F(n + 1) - 5125/4 F(n) F(n + 1) 28 6351 3 3 5 2 435 2 + ---- F(n) F(n + 1) + 2165/2 F(n) F(n + 1) + 6/7 F(n) + --- F(n + 1) 28 28 69 14557 4 - -- F(n) F(n + 1) - ----- F(n + 1) 14 28 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 559 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(3 n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 7 6 G(n) = 7/44 F(n + 1) - 1/11 F(n + 1) + 3/44 F(n + 1) - 3/11 F(n) 7 147 5 2 12 2 4 - 8/11 F(n) - --- F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 109 2 5 95 3 4 31 4 2 + --- F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 44 22 22 30 6 3 3 355 4 3 + -- F(n) F(n + 1) - 5/11 F(n) F(n + 1) + --- F(n) F(n + 1) 11 44 37 5 6 + -- F(n) F(n + 1) - 3/22 F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 560 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(3 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 172 4 17375 6 39410 2 5 G(n) = ---- F(n) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 319 45960 4 3 17810 2 290 3 5 20 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + --- F(n) F(n + 1) + --- 319 319 11 319 805 3 20 3 11290 3 4 1845 4 4 - --- F(n) - --- F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 11 1754 3 7762 2 2 1780 2 6 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 11 22492 3 760 7 5183 2 + ----- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 11 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 561 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(3 n - j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 316 3 128 2 4539 4 G(n) = -415/2 F(n + 1) - --- F(n + 1) + --- F(n) F(n + 1) + ---- F(n + 1) 11 11 22 135 3 2 6 3 5 - --- F(n) F(n + 1) - 5/2 F(n) F(n + 1) - 435 F(n) F(n + 1) 11 3 4 2 5 763 2 2 + 121/2 F(n) F(n + 1) - 3 F(n) F(n + 1) + --- F(n) F(n + 1) 22 81 2 7 6 - -- F(n) F(n + 1) + 505 F(n) F(n + 1) - 69 F(n) F(n + 1) 22 1913 3 31 13 3 7 - ---- F(n) F(n + 1) + -- + -- F(n) + 57/2 F(n + 1) 22 22 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 562 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(3 n - j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 1157 3 2 4208 4 19 94 3 G(n) = ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - -- + -- F(n) F(n + 1) 11 11 22 11 2 6 9820 5 9 47 + 15/2 F(n) F(n + 1) + ---- F(n + 1) - 1825/2 F(n + 1) - -- F(n) 11 22 519 2 2 4281 2 3 + 20 F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 22 22 3 5 3 6 1217 3 + 260 F(n) F(n + 1) - 3675/2 F(n) F(n + 1) + ---- F(n) F(n + 1) 22 7 8 2 7 - 625/2 F(n) F(n + 1) + 2250 F(n) F(n + 1) - 225/2 F(n) F(n + 1) 8 2791 4 + 255/2 F(n + 1) - ---- F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 563 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5747 3 8 23823 10 G(n) = ----- F(n) F(n + 1) + ------- F(n) F(n + 1) 83182 2412278 3235428 3 7 812452 2 - ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 1206139 1206139 231102 9 22724148 5 2 503145 2 + ------- F(n) F(n + 1) - -------- F(n) F(n + 1) + ------- F(n + 1) 1206139 6030695 2412278 924408 10 5310 11 929718 2 503145 3 + ------- F(n) + ------- F(n) - ------- F(n) - ------- F(n + 1) 1206139 1206139 1206139 2412278 19181466 5 9548953 4 7 + -------- F(n) F(n + 1) + -------- F(n) F(n + 1) 6030695 12061390 24496812 4 6 2659869 4 3 + -------- F(n) F(n + 1) - ------- F(n) F(n + 1) 6030695 6030695 6008652 4 2 14063 10 - ------- F(n) F(n + 1) + ----- F(n) F(n + 1) 6030695 7562 1115579 9 2 462204 9 + ------- F(n) F(n + 1) + ------- F(n) F(n + 1) 634810 6030695 28869337 8 3 3697632 8 2 + -------- F(n) F(n + 1) - ------- F(n) F(n + 1) 2412278 1206139 14189302 7 4 693306 7 3 - -------- F(n) F(n + 1) - ------- F(n) F(n + 1) 1206139 1206139 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 564 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 4 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7773 3 3764615 2 5 153500 2 9 G(n) = ------ F(n) F(n + 1) + ------- F(n) F(n + 1) - ------ F(n) F(n + 1) 12122 12122 551 1143 2 2 3975195 3 4 2300625 3 8 + ---- F(n) F(n + 1) - ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 6061 12122 1102 3016125 10 4031495 6 2997 3 + ------- F(n) F(n + 1) - ------- F(n) F(n + 1) - ---- F(n) F(n + 1) 1102 12122 6061 285793 2 46491 2 27 4 - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n + 1) 12122 12122 12122 321 3 321 1203625 11 6026795 7 - ---- F(n) + ---- - ------- F(n + 1) + ------- F(n + 1) 6061 6061 1102 6061 592835 3 + ------ F(n + 1) 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 565 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 4 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 4 6 G(n) = 1/84 F(n) (-214 F(n) F(n + 1) + 79508 F(n) F(n + 1) 4 4 4 2 8 - 20 F(n) F(n + 1) - 101 F(n) + 101 F(n) - 20 F(n + 1) 5 3 - 37556 F(n) F(n + 1) + 40 F(n) F(n + 1) - 718 F(n) F(n + 1) 10 4 2 3 7 + 8 F(n + 1) + 3603 F(n) F(n + 1) + 1891 F(n) F(n + 1) 3 5 3 3 3 - 40 F(n) F(n + 1) - 6999 F(n) F(n + 1) + 194 F(n) F(n + 1) 2 8 2 6 2 4 - 93383 F(n) F(n + 1) + 20 F(n) F(n + 1) + 15412 F(n) F(n + 1) 9 7 + 38234 F(n) F(n + 1) + 40 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 566 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 4 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 156 3 9546 2 183 2 2 G(n) = --- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 209 6061 418 15075 2 5 549 3 41625 6 + ----- F(n) F(n + 1) - --- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 418 1102 427725 3 4 2868 219 4 2868 3 - ------ F(n) F(n + 1) + ---- - --- F(n + 1) - ---- F(n) 12122 6061 418 6061 96270 3 36045 2 191925 7 + ----- F(n + 1) - ----- F(n) F(n + 1) - ------ F(n + 1) 6061 6061 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 567 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 4 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 4 G(n) = -1/924 F(n) (-2011 F(n) F(n + 1) + 755 F(n) + 5325 F(n) F(n + 1) 2 4 3 3 6 - 11200 F(n) F(n + 1) + 46400 F(n) F(n + 1) + 17600 F(n + 1) 5 2 2 4 - 40000 F(n) F(n + 1) + 2710 F(n) F(n + 1) + 699 F(n + 1) 3 2 2 - 1398 F(n) F(n + 1) - 18125 F(n + 1) - 755 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 568 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 4 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 813771 7 1646478 2 503145 3 503145 2 G(n) = -------- F(n) - ------- F(n) + ------- F(n + 1) - ------- F(n + 1) 1206139 6030695 2412278 2412278 6071676 300807 6 367461 5 + ------- F(n) F(n + 1) + ------ F(n) - ------ F(n) F(n + 1) 6030695 317405 126962 73575 5 2 8501661 6 7454637 2 - ----- F(n) F(n + 1) + ------- F(n) F(n + 1) + ------- F(n) F(n + 1) 7562 2412278 2412278 34509 5 212049 2 3742191 6 - ----- F(n) F(n + 1) + ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 28855 2412278 1206139 69093 2 4 10455867 2 5 + ------ F(n) F(n + 1) + -------- F(n) F(n + 1) 126962 1206139 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 569 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 69 4 83480 2 2 669 2 3 G(n) = --- F(n) F(n + 1) + ----- F(n) F(n + 1) - --- F(n) F(n + 1) 209 6061 418 50735 3 21 3 2 1101 4 - ----- F(n) F(n + 1) - -- F(n) F(n + 1) + ---- F(n) F(n + 1) 12122 19 418 99025 3 5 615 4887 615 9159 4 - ----- F(n) F(n + 1) + ----- - ---- F(n) - ----- F(n + 1) + ----- F(n) 12122 12122 6061 12122 12122 460517 3 232900 7 547150 2 6 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 12122 6061 6061 534650 4 4 + ------ F(n) F(n + 1) 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 570 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 4 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2903 2 863 4 2903 6 2725 8 G(n) = ---- F(n + 1) + --- F(n + 1) - ---- F(n + 1) - ---- F(n + 1) 132 66 132 231 197 2 5 1796 3 3 + --- F(n) + 301/6 F(n) F(n + 1) - ---- F(n) F(n + 1) 154 33 5587 3 299 1525 7 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 462 42 21 7269 3 18400 2 6 97 2 4 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) + -- F(n) F(n + 1) 77 77 11 2 2 18050 4 4 197 - 260/7 F(n) F(n + 1) - ----- F(n) F(n + 1) - --- 77 154 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 571 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 4 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 105 17631 5 9777 3 6 G(n) = ----- F(n) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 30305 12122 26262 4 366 4 2 77409 4 5 + ----- F(n) F(n + 1) + --- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 209 12122 38901 5 675 7 2 269721 8 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) + ------ F(n) F(n + 1) 12122 22 12122 981 28173 6 615 6 67504 5 + ----- F(n) F(n + 1) + ----- F(n) - ----- F(n + 1) - ----- F(n) 30305 30305 12122 6061 59854 9 615 5 6381 4 558 2 + ----- F(n) + ----- F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) 6061 12122 12122 1595 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 572 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 303 4 13315 11 959609 8 3 G(n) = ----- F(n + 1) - ----- F(n) + ------ F(n) F(n + 1) 1102 12122 12122 16431 11 92246 6 529 3 + ----- F(n + 1) - ----- F(n) F(n + 1) + --- F(n) F(n + 1) 1045 2755 551 174964 10 259367 2 5 397 2 2 - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) - --- F(n) F(n + 1) 6061 6061 551 199765 9 2 541513 10 201449 2 + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 6061 60610 30305 12096 2 1193 554751 3 907 3 - ----- F(n) F(n + 1) + ----- - ------ F(n + 1) - ---- F(n) F(n + 1) 1595 12122 30305 1102 83602 7 + ----- F(n + 1) 30305 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 573 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 3 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 75 9 37 5 29 11 2311 8 3 G(n) = 5/56 F(n + 1) - -- F(n) + -- F(n) + -- F(n) - ---- F(n) F(n + 1) 28 14 28 42 8 13 2 5 25 2 3 - 45/7 F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 24 56 1097 10 4255 9 2 2413 5 2 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 60 84 420 403 4 3 59 4 13 3 8 - --- F(n) F(n + 1) + -- F(n) F(n + 1) + -- F(n) F(n + 1) 168 56 42 15 3 6 75 6 3 2356 6 - -- F(n) F(n + 1) + -- F(n) F(n + 1) - ---- F(n) F(n + 1) 28 14 105 1817 5 6 13 7 13 3 + ---- F(n) F(n + 1) - 5/56 F(n + 1) - --- F(n + 1) + --- F(n + 1) 420 168 168 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 574 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 3 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 11 19465 3 4 349 2 289 3 G(n) = ---- F(n + 1) - ----- F(n) F(n + 1) - --- F(n) - --- F(n) 319 58 319 319 100930 2 9 20 4 6 79180 4 3 - ------ F(n) F(n + 1) + --- F(n) F(n + 1) + ----- F(n) F(n + 1) 29 319 319 2253 4 2 78335 3 8 40 3 7 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 878 3 3 447540 10 40 9 - --- F(n) F(n + 1) + ------ F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 319 20 2 8 11895 2 5 2883 2 4 - --- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 22 319 3513 2 29515 6 1105 5 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 22 319 19353 2 1897 884605 4 7 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) 319 638 319 20 10 + --- F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 575 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 17485 3 252 3 17525 7 67 19 3 G(n) = ----- F(n + 1) - --- F(n) - ----- F(n + 1) - --- - -- F(n) F(n + 1) 638 319 638 319 22 1857 2 2 2 2725 2 5 + ---- F(n) F(n + 1) - 2/11 F(n) F(n + 1) + ---- F(n) F(n + 1) 638 638 13 3 4 1900 6 + -- F(n) F(n + 1) + 3/11 F(n + 1) + ---- F(n) F(n + 1) 22 29 39075 3 4 6513 2 - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 576 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 3 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 3 6 G(n) = 1/84 F(n) (-549 F(n) F(n + 1) + 66 F(n) F(n + 1) + 49 F(n + 1) 3 2 2 - 424 F(n) F(n + 1) + 102 F(n) F(n + 1) - 201 F(n) F(n + 1) 3 3 4 2 2 4 - 777 F(n) F(n + 1) + 1424 F(n) F(n + 1) + 51 F(n) + 33 F(n) 5 4 + 277 F(n) F(n + 1) - 51 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 577 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 3 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4695 7 1070 7 1070 6 47493 6 G(n) = ----- F(n) + ---- F(n + 1) - ---- F(n + 1) + ----- F(n) F(n + 1) 6061 6061 6061 12122 7427 6 1219 5 173 6 - ---- F(n) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 6061 1102 551 832 5 20665 3 4 42835 4 3 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 6061 6061 14955 2 5 84465 5 2 248 3 3 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - --- F(n) F(n + 1) 6061 12122 319 543 2 4 15 4 2 - --- F(n) F(n + 1) + --- F(n) F(n + 1) 418 209 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 578 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 200 2 6 1543 3 79 4 G(n) = --- F(n) F(n + 1) - ---- F(n) F(n + 1) - -- F(n) F(n + 1) 319 319 22 4121 2 2 2 3 61 3 2 351 + ---- F(n) F(n + 1) - 1/22 F(n) F(n + 1) + -- F(n) F(n + 1) + --- 319 22 638 53425 8 29 5 287 801 - ----- F(n + 1) + -- F(n + 1) + --- F(n) - --- F(n + 1) 638 22 638 638 26517 4 10871 3 64825 7 + ----- F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 319 56600 3 5 - ----- F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 579 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 19 4 6 2 29 3 G(n) = 16/7 F(n + 1) - -- F(n) - 1/7 F(n) - 1/2 F(n) + -- F(n) F(n + 1) 14 24 6 2 1745 5 3 4 2 - 61/2 F(n) F(n + 1) + ---- F(n) F(n + 1) + 6 F(n) F(n + 1) 84 155 3 5 3 3 1139 3 + --- F(n) F(n + 1) - 10/7 F(n) F(n + 1) + ---- F(n) F(n + 1) 84 168 29 7 5 7 - -- F(n) F(n + 1) - 37/7 F(n) F(n + 1) + 25/8 F(n) F(n + 1) 56 2 - 16/7 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 580 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 98045 3 98375 7 86419 2 G(n) = ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 1102 1102 12122 18404 2 2 229275 2 6 10187 3 + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 551 1102 1102 13 3 4 227325 4 4 22575 3 5 + -- F(n) F(n + 1) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 22 1102 1102 23787 3 1070 3 1070 949 4 100087 2 + ----- F(n) - ---- F(n + 1) + ---- + ---- F(n) + ------ F(n) F(n + 1) 12122 6061 6061 1102 6061 183 6 829 2 5 793 4 3 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 11 22 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 581 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 855 20 9 73 2 4 3 3 G(n) = ---- F(n) - --- F(n + 1) + -- F(n) F(n + 1) - 2/11 F(n) F(n + 1) 638 319 11 803 3 2 45631 2 3 244285 4 5 - --- F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 29 638 638 559 4 3705 3 6 10355 8 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 58 638 58 112495 4 71 5 65 4 2 - ------ F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 638 22 11 421 2 809 274735 2 7 20 2 - --- F(n) + --- F(n) F(n + 1) - ------ F(n) F(n + 1) + --- F(n + 1) 638 319 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 582 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 4 2 G(n) = -1/168 F(n) (1433 + 2184 F(n + 1) - 1763 F(n) - 174 F(n) 4 2 - 78567 F(n + 1) - 2046 F(n + 1) + 198 F(n) F(n + 1) 7 5 3 3 - 186200 F(n) F(n + 1) - 4704 F(n) F(n + 1) + 2016 F(n) F(n + 1) 2 6 3 5 4 2 - 2800 F(n) F(n + 1) + 165200 F(n) F(n + 1) + 2856 F(n) F(n + 1) 2 2 3 8 - 9967 F(n) F(n + 1) + 35334 F(n) F(n + 1) + 77000 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 583 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 10 20 3 4187 2 G(n) = ---- F(n + 1) + --- F(n + 1) + ---- F(n) F(n + 1) 319 319 3190 48405 6 4 3417 7 3 4341 9 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 116 232 1319 5 3059 9 1003 2 5 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 580 2552 319 3719 2 5989 2 354 7 41 6 + ---- F(n) F(n + 1) + ---- F(n) + --- F(n) + ---- F(n) 3190 3190 319 3190 97051 5 5 23 5 2 6987 6 - ----- F(n) F(n + 1) - -- F(n) F(n + 1) - ---- F(n) F(n + 1) 2552 11 3190 11509 6 5447 8 2 2473 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 3190 638 1160 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 584 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 27 2 19 327 2 4 G(n) = --- F(n) + -- F(n) F(n + 1) - --- F(n) F(n + 1) 638 29 319 1690875 3 8 1076 3 3 6939 2 - ------- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 638 2216750 10 539215 6 19527 2 + ------- F(n) F(n + 1) - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 319 638 319 530835 3 4 665 3 225625 2 9 - ------ F(n) F(n + 1) - --- F(n) - ------ F(n) F(n + 1) 638 638 319 503315 2 5 1171 2 158215 3 + ------ F(n) F(n + 1) - ---- F(n + 1) + ------ F(n + 1) 638 638 638 1464 5 678 6 805425 7 - ---- F(n) F(n + 1) + --- F(n + 1) + ------ F(n + 1) 319 319 319 884625 11 - ------ F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 585 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 725 7 2 3 2 5 G(n) = --- F(n + 1) + 5/3 F(n) F(n + 1) - 25/3 F(n) F(n + 1) 12 3 2 1625 3 4 6 - 8/3 F(n) F(n + 1) + ---- F(n) F(n + 1) - 575/4 F(n) F(n + 1) 12 13 3 725 3 2 2 + -- F(n) - --- F(n + 1) - 41/6 F(n) F(n + 1) + 47/2 F(n) F(n + 1) 12 12 5 - 1/3 F(n + 1) - 1/12 F(n) + 1/3 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 586 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 2 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 359 3 427 3 147 2 107 2 5 G(n) = ---- F(n) + ---- F(n + 1) - --- F(n + 1) + --- F(n) F(n + 1) 638 1276 638 44 113 3 4 549 4 2 2032 5 2 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 58 638 319 523 2 4 40 3 3 4947 4 3 - --- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 1276 961 5 1317 6 57 7 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n + 1) 638 319 1276 19 6 279 6 - --- F(n + 1) - --- F(n) 319 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 587 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1721565 3 78225 7 30531 3 G(n) = ------- F(n + 1) - ----- F(n + 1) - ----- F(n) 12122 551 12122 335238 2 5835 2909 3 65 2 6 - ------ F(n) F(n + 1) - ----- + ---- F(n) F(n + 1) - -- F(n) F(n + 1) 6061 12122 418 11 97155 2 5257 2 2 4450 7 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 209 11 175350 3 4 3980 3 5 14539 3 - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 551 11 209 372225 6 21525 2 5 34965 4 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n + 1) 1102 1102 209 1845 8 + ---- F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 588 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 4 7 G(n) = -1/12 F(n) (14 F(n) - 30 F(n) + 68 F(n) F(n + 1) - 10 F(n + 1) 6 5 2 5 3 2 + 35 F(n) F(n + 1) + 391 F(n) F(n + 1) + 28 F(n) - 60 F(n) F(n + 1) 6 2 2 3 + 23 F(n) F(n + 1) - 103 F(n) F(n + 1) - 4 F(n) F(n + 1) 5 3 4 4 3 + 12 F(n + 1) - 36 F(n) F(n + 1) - 328 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 589 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6000 3 5 3852 2 134275 4 4 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) 319 319 638 1863 3 4 68500 2 6 3119 3 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 319 319 1434 2 11092 2 2 58361 3 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 638 29275 7 635 4 228 3 185 + ----- F(n) F(n + 1) + --- F(n) + --- F(n) + --- 319 638 319 638 9184 4 3 8227 2 5 3643 6 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 185 7 - --- F(n + 1) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 590 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 1723 6 3 3389 7 2 457 3 6 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 32 48 24 1523 4 4 3 389 4 5 - ---- F(n) F(n + 1) + 77/6 F(n) F(n + 1) - --- F(n) F(n + 1) 96 24 5 2 179 5 4 13 6 - 35/6 F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) F(n + 1) 48 12 65 9 2 39 53 3 - -- F(n + 1) + 1/3 F(n) F(n + 1) - -- F(n + 1) - -- F(n + 1) 96 32 12 7 53 7 5 91 5 125 6 - 1/4 F(n) + -- F(n + 1) + 9/4 F(n) + -- F(n + 1) - --- F(n) F(n + 1) 12 48 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 591 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6197 2 2 45541 2 3 2 7 G(n) = ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - 2475/2 F(n) F(n + 1) 116 116 3807 3 311529 3 2 3 6 + ---- F(n) F(n + 1) - ------ F(n) F(n + 1) + 825/2 F(n) F(n + 1) 58 638 22899 4 3 5 4 5 - ----- F(n) F(n + 1) - 105/2 F(n) F(n + 1) + 1125 F(n) F(n + 1) 116 4 4 2 6 271455 - 625/4 F(n) F(n + 1) + 655/4 F(n) F(n + 1) + ------ F(n + 1) 1276 9 36857 1663 4 3217 4 542235 5 + 425/2 F(n + 1) - ----- + ---- F(n + 1) + ---- F(n) - ------ F(n + 1) 1276 29 116 1276 8 1179 5 - 115/4 F(n + 1) - ---- F(n) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 592 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 6 3 230351 2 G(n) = 65/3 F(n) F(n + 1) - 15/4 F(n) F(n + 1) - ------ F(n + 1) 2040 724259 2 2823397 6 682439 10 1441291 10 - ------ F(n) + ------- F(n + 1) + ------ F(n) - ------- F(n + 1) 12240 12240 12240 12240 8 7 85355 9 + 5/12 - 425/3 F(n + 1) + 1010/3 F(n) F(n + 1) + ----- F(n) F(n + 1) 204 2 2 376169 2 4 919013 2 8 + 16 F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 2448 2448 3 5 266317 9 255143 - 320 F(n) F(n + 1) + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 1224 6120 343556 5 3 4 - ------ F(n) F(n + 1) - 54 F(n) F(n + 1) + 565/4 F(n + 1) 765 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 593 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 76 3 4 3 G(n) = -2/5 F(n + 1) - -- F(n) F(n + 1) - 331/4 F(n) F(n + 1) 55 17 2 22781 5 2 2425 4 7 - -- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 30 240 24 4 4 114 5 3 6 5 - 5/22 F(n) F(n + 1) - --- F(n) F(n + 1) + 1037/4 F(n) F(n + 1) 55 70 7 15113 5 6 1225 6 + -- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 48 24 62 6 2 21 13 3 62 4 37 7 - -- F(n) F(n + 1) - -- - -- F(n + 1) + -- F(n + 1) + -- F(n + 1) 11 22 40 55 60 17 11 19 8 1573 7 19367 11 - --- F(n + 1) + -- F(n) + ---- F(n) - ----- F(n) 264 22 48 528 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 594 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 21 3 23 2 17 5 30 2 4 G(n) = --- F(n) - -- F(n) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 22 11 11 303 2 63 4 2 1305 2 5 - --- F(n) F(n + 1) - -- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 11 22 90 2 295 6 47 + -- F(n) F(n + 1) + --- F(n) F(n + 1) + -- F(n) F(n + 1) 11 11 22 1555 4 3 170 3 4 23 3 3 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 22 6 7 - 5/22 F(n + 1) + 5/22 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 595 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 187975 6 91710 3 4 39955 7 G(n) = ------ F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n + 1) 638 319 319 195 3 14481 2 1495 3 + --- F(n) F(n + 1) - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 22 319 22 4450 7 567 2 2 8609 2 5 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 22 319 3980 3 5 7461 2 39935 3 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n + 1) 11 638 319 65 2 6 743 1845 8 1809 7 4 - -- F(n) F(n + 1) - --- + ---- F(n + 1) - ---- F(n) - 333/2 F(n + 1) 11 638 11 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 596 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2175 7 9 31 3 2175 3 G(n) = ----- F(n + 1) + 2275 F(n + 1) - -- F(n) + ---- F(n + 1) 28 21 28 5175 6 3 6 4875 3 4 + ---- F(n) F(n + 1) + 18125/4 F(n) F(n + 1) - ---- F(n) F(n + 1) 28 28 2 3 731 2 8 - 3109/6 F(n) F(n + 1) + --- F(n) F(n + 1) - 22525/4 F(n) F(n + 1) 84 11389 4 2531 2 3 2 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + 1841/6 F(n) F(n + 1) 12 84 2 5 2 7 157 + 75/7 F(n) F(n + 1) + 1425/4 F(n) F(n + 1) + --- F(n) 21 5 - 251/4 F(n + 1) - 8849/4 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 597 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 67 3 4 31 7 2 G(n) = -- F(n) F(n + 1) - -- F(n) F(n + 1) + 1/2 F(n) F(n + 1) 22 11 221 2 2 127 4 3 3 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - 7/11 F(n) F(n + 1) 22 22 43 2 5 8 26 7 7 - -- F(n) F(n + 1) + 5/22 F(n + 1) + -- F(n) - 5/22 F(n + 1) 22 11 551 3 5 71 3 8 13 3 - --- F(n) F(n + 1) - -- F(n) F(n + 1) + 9/11 F(n) - -- F(n) 22 22 11 675 6 2 6 289 2 6 + --- F(n) F(n + 1) + 3/22 F(n) F(n + 1) + --- F(n) F(n + 1) 22 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 598 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5714 4 5 34721 4 20 1029 3 G(n) = ----- F(n) F(n + 1) + ----- F(n) - --- - ---- F(n) F(n + 1) 319 1276 319 638 29589 8 91755 6 2 17140 6 3 - ----- F(n) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 1276 1276 319 23945 4 4 10645 3 6 8130 4 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 1276 638 319 15911 3 5 1731 2 6 97 2 7 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 638 638 319 23994 7 22184 7 2 4120 8 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 319 20 9 479 9 823 + --- F(n + 1) + --- F(n) + --- F(n) 319 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 599 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 39 177 3 49 4 9 G(n) = --- F(n) + --- F(n) F(n + 1) - -- F(n) + 5/22 F(n + 1) 22 22 22 10040 4 5 8 3881 2 3 + ----- F(n) F(n + 1) - 5/22 F(n + 1) + ---- F(n) F(n + 1) 11 22 4670 8 1255 7 9313 4 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 22 22 632 3 523 2 2 2975 2 6 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 22 22 22555 2 7 725 3 2 221 4 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 22 11 11 50 3 5 265 3 6 1405 4 4 + -- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 22 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 600 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 n - j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 955 5 9300 30100 445857 6 G(n) = ---- F(n + 1) - ---- F(n) + ----- F(n + 1) - ------ F(n + 1) 142 781 781 3476 126113 10 2175 9 135119 2 446647 10 - ------ F(n) + ---- F(n) + ------ F(n) + ------ F(n + 1) 3476 142 3476 3476 389403 3 3 8480 3 2 13675 2 7 + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 1738 71 142 650821 2 4 3607 2 3 389100 9 - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 3476 71 869 15325 8 338137 5 3196 4 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 142 1738 71 301495 9 3292 1389155 2 8 - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) + ------- F(n) F(n + 1) 1738 869 3476 2275 9 - ---- F(n + 1) 71 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 601 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(3 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 27 2 10 3 G(n) = - 5/22 + 5/22 F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 21 2 13 2 2 23 3 3 + -- F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) + 4/11 F(n) F(n + 1) 11 11 22 4 + 3/11 F(n) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 602 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = 1/12 F(n) (5 - 3 F(n + 1) + 3 F(n) F(n + 1) - 2 F(n) 2 2 3 4 + 27 F(n) F(n + 1) - 39 F(n) F(n + 1) + 9 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 603 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5983 153 2 4 197 3 835 2 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) - --- F(n) 638 58 319 319 1589 5 1304 3 3 651 2 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 29 29 638 531 2 14205 2 1295 6 20 3 + --- F(n) F(n + 1) - ----- F(n + 1) + ---- F(n + 1) - --- F(n + 1) 638 638 58 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 604 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 27 12 3 65 2 2 G(n) = 5/22 - 5/22 F(n + 1) + -- F(n) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 22 2 3 36 3 30 3 2 + 4/11 F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 11 11 4 27 4 12 5 + 7/11 F(n) - -- F(n) F(n + 1) - -- F(n) 11 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 605 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 23 2 2 2 4 G(n) = 1/12 F(n + 1) - -- F(n) - 25 F(n + 1) - 1/12 - 27/4 F(n) F(n + 1) 12 3 2 2 103 + 1/4 F(n) F(n + 1) + 1/6 F(n) F(n + 1) + --- F(n) F(n + 1) 12 3 6 3 3 - 1/3 F(n) F(n + 1) + 25 F(n + 1) + 233/4 F(n) F(n + 1) 5 - 233/4 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 606 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5517 7 767 3 11555 2 G(n) = ---- F(n + 1) - --- F(n) F(n + 1) - ----- F(n) F(n + 1) 58 638 638 1114 2 2 1079 2 5 3257 3 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 58 638 1327 3 60647 3 350 4 28747 2 + ---- F(n) - ----- F(n + 1) + --- F(n + 1) + ----- F(n) F(n + 1) 319 638 319 638 6854 6 10801 3 4 370 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - --- 29 58 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 607 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 45 2 3 21 5 6 5 G(n) = -- F(n) F(n + 1) - -- F(n) - 1/11 F(n) + 5/22 F(n + 1) 11 11 6 73 2 4 73 3 2 - 5/22 F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 49 3 3 119 4 57 4 2 + -- F(n) F(n + 1) + --- F(n) F(n + 1) - -- F(n) F(n + 1) 11 22 22 5 41 4 5 + 5/11 F(n) F(n + 1) - -- F(n) F(n + 1) + 2 F(n) F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 608 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 G(n) = -1/12 F(n) (192 F(n) F(n + 1) + F(n + 1) - 2 F(n) F(n + 1) 2 2 2 4 3 6 - F(n) F(n + 1) - 231 F(n) F(n + 1) + 2 F(n) F(n + 1) + 582 F(n + 1) 5 3 3 2 2 - 1353 F(n) F(n + 1) + 1431 F(n) F(n + 1) - 585 F(n + 1) - 37 F(n) 4 + F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 609 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 2 4 4 2 G(n) = 15 F(n) F(n + 1) - 30 F(n) F(n + 1) + 105/2 F(n) F(n + 1) 184 13 2 54 6 - --- F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 11 11 11 172 3 4 3 3 101 2 5 + --- F(n) F(n + 1) - 25 F(n) F(n + 1) + --- F(n) F(n + 1) 11 11 267 4 3 3 61 2 149 2 - --- F(n) F(n + 1) + 5/22 F(n) + -- F(n) + --- F(n) F(n + 1) 11 22 22 3 2 - 5/22 F(n + 1) + 5/22 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 610 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 369 2 183 2 210 5 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 638 319 2252 5 1631 2 4 1330 3 3 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 319 931 4 2 307 3 185 3 185 6 - --- F(n) F(n + 1) + --- F(n) - --- F(n + 1) + --- F(n + 1) 319 638 638 638 945 2 - --- F(n) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 611 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 12615 8 615 3 42447 6081 3 G(n) = ----- F(n + 1) - ----- F(n + 1) - ----- + ----- F(n) 38 12122 12122 12122 3981123 4 8199 2 14944 3 - ------- F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 12122 12122 551 16965 2 6 140090 3 5 1776881 3 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + ------- F(n) F(n + 1) 418 209 12122 341995 7 5022 2 809235 2 2 - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 418 6061 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 612 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 3 2 5 G(n) = -1/24 F(n) (-210 F(n) F(n + 1) + 71 F(n) F(n + 1) - 16 F(n + 1) 2 4 2 - 18 F(n) F(n + 1) + 103 F(n) F(n + 1) + 34 F(n) F(n + 1) 3 5 3 + 12 F(n + 1) + 101 F(n) - 14 F(n) - 63 F(n)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 613 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 11111 2 185 3 834 3 1018 6 G(n) = ----- F(n) F(n + 1) + --- F(n + 1) + --- F(n) - ---- F(n) F(n + 1) 319 638 319 29 2457 2 5 4473 4 3 185 63 3 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- - --- F(n) F(n + 1) 29 58 638 638 4052 2 1325 2 2 1451 3 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 638 638 173 3 4 207 4 + --- F(n) F(n + 1) - --- F(n) 58 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 614 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 6 2 G(n) = -1/12 F(n) (-1287 F(n) F(n + 1) + 558 F(n + 1) - 569 F(n + 1) 3 2 2 3 - 18 F(n) F(n + 1) + 41 F(n) F(n + 1) - 32 F(n) F(n + 1) 3 3 4 4 2 4 + 1389 F(n) F(n + 1) + 9 F(n + 1) + 9 F(n) - 249 F(n) F(n + 1) 2 + 182 F(n) F(n + 1) - 33 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 615 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 117 4 699 1847 11215 2 6 G(n) = ---- F(n + 1) + --- F(n) - ---- F(n + 1) + ----- F(n) F(n + 1) 638 638 319 29 3 2 48223 3 20225 3 + 33/2 F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 638 2112 2 2 2 3 9480 4 4 - ---- F(n) F(n + 1) - 13/2 F(n) F(n + 1) - ---- F(n) F(n + 1) 29 29 4 2613 265 8 4900 7 - 12 F(n) F(n + 1) - ---- + --- F(n + 1) - ---- F(n) F(n + 1) 638 58 29 5 + 11/2 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 616 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 5 4 3 G(n) = -1/60 F(n) (260 F(n) - 10 F(n) - 70 F(n) - 2620 F(n) F(n + 1) 5 7 2 + 320 F(n + 1) - 719 F(n + 1) - 260 F(n + 1) + 8 F(n) F(n + 1) 3 6 6 + 649 F(n + 1) + 1535 F(n) F(n + 1) - 1130 F(n) F(n + 1) 5 2 4 4 + 1937 F(n) F(n + 1) + 700 F(n) F(n + 1) - 600 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 617 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1070 3 3339 2 3296 3 19878 4 1070 G(n) = ---- F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) - ----- F(n) - ---- 6061 6061 6061 6061 6061 1325 3 5 12615 4 4 123526 3 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 209 38 6061 77865 2 6 780499 2 2 6549 2 + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 209 12122 6061 64465 7 1874025 3 - ----- F(n) F(n + 1) + ------- F(n) F(n + 1) 418 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 618 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 4 6 2 G(n) = -1/1848 F(n) (-24045 F(n) F(n + 1) - 136383 F(n) F(n + 1) 3 5 8 7 + 28966 F(n) F(n + 1) - 270 F(n + 1) + 141740 F(n) F(n + 1) 2 3 7 8 - 78 F(n + 1) + 3094 F(n) F(n + 1) - 2014 F(n) F(n + 1) + 36347 F(n) 4 2 - 47383 F(n) - 52 F(n) + 78 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 619 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 49163 3 1407 3 131 1097 2 2 G(n) = ------ F(n + 1) + ---- F(n) - --- - ---- F(n) F(n + 1) 638 638 638 638 4473 7 9119 3 4 5491 6 + ---- F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 58 58 29 367 3 95 3 3791 2 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 91 4 10995 2 441 2 5 + --- F(n + 1) + ----- F(n) F(n + 1) + --- F(n) F(n + 1) 638 319 58 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 620 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7058 24595 5 53 8 1070 6570 4 5 G(n) = ---- F(n) + ----- F(n) + ---- F(n) + ---- - ---- F(n) F(n + 1) 6061 1102 6061 6061 6061 19333 3 2 1553 2 2 17954 2 3 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 1102 6061 106 5 3 1707 3 53 6 2 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 6061 6061 53 4 4 15765 3 896815 8 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 6061 12122 12122 746445 7 2 106 7 10220 6 3 + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 12122 6061 6061 1070 9 214175 9 - ---- F(n + 1) - ------ F(n) 6061 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 621 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5633 53 5 73 5 544 2017 G(n) = ---- + --- F(n) - -- F(n + 1) + ---- F(n) + ---- F(n + 1) 5104 110 55 1595 1595 62299 4 23137 8 4325 8 4819 4 + ----- F(n) - ----- F(n) + ---- F(n + 1) - ---- F(n + 1) 5104 1276 5104 2552 38655 6 2 39345 7 4 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - 7/2 F(n) F(n + 1) 5104 2552 61 4 16991 3 5 107535 5 3 + -- F(n) F(n + 1) - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 22 1276 2552 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 622 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 328 6 44612 5 1200 5 1671 5 G(n) = ---- F(n) + ----- F(n) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 4785 319 319 1555 6 10982 18451 - 9/319 F(n) F(n + 1) - ---- F(n + 1) - ----- F(n) - ----- F(n + 1) 638 4785 12760 1515 2 10865 4 5 90 4 2 + ---- F(n + 1) + ----- F(n) F(n + 1) - -- F(n) F(n + 1) 638 174 11 278243 4 47185 7 2 43075 6 3 - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 7656 348 696 37093 5 4 343 9 1321 5 - ----- F(n) F(n + 1) - --- F(n + 1) + ---- F(n + 1) 348 696 660 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 623 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 95839350 2 8 1852543758 3 3 G(n) = -------- F(n) F(n + 1) + ---------- F(n) F(n + 1) 63481 1206139 744909450 3 7 374314029 2 4 + --------- F(n) F(n + 1) - --------- F(n) F(n + 1) 63481 219298 772551 2 972049800 9 - ------- F(n) F(n + 1) - --------- F(n) F(n + 1) 1206139 63481 2611161387 5 101350983 + ---------- F(n) F(n + 1) + --------- F(n) F(n + 1) 1206139 1206139 6877278545 6 388305475 10 22237425 2 - ---------- F(n + 1) + --------- F(n + 1) - -------- F(n) 1206139 63481 2412278 1000547815 2 526923 3 503145 3 - ---------- F(n + 1) + ------- F(n) - ------- F(n + 1) 2412278 2412278 2412278 35667 2 + ------- F(n) F(n + 1) 2412278 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 624 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 23475 3 615 9 7485 4 36345 9 G(n) = ----- F(n) F(n + 1) + ----- F(n + 1) - ----- F(n) + ----- F(n) 24244 12122 12122 6061 4053 3 5 27149 3 6 45579 4 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 1276 12122 12122 109575 5 3 90910 5 4 13773 6 2 - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 24244 6061 12122 49243 6 3 20355 7 312570 7 2 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 1102 12122 6061 254233 8 12313 4 13773 7 - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 12122 12122 24244 289 8 615 4071 8 + --- F(n) F(n + 1) - ----- + ---- F(n) 638 12122 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 625 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2879 10 32 44306 7 3 38047 8 2 G(n) = ----- F(n) + --- + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 308 231 231 231 1671 9 151 2 4 2117 4 4 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 154 154 924 281 4 2 17135 5 12878 6 4 - --- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 22 462 231 599 6 2 151 2 16 4 151 6 - --- F(n) F(n + 1) - --- F(n + 1) - -- F(n + 1) + --- F(n + 1) 308 924 77 924 48 7 641 3 3 267 3 5 - -- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 77 154 154 16 8 193 8 16 5 3 + --- F(n + 1) + --- F(n) + -- F(n) F(n + 1) 231 924 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 626 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(4 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 16431 5 4 33501 2 7 3963 G(n) = ------ F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n + 1) 6061 24244 24244 1737 2 549 9 10467 10 848648 7 3 + ----- F(n + 1) + ---- F(n) - ----- F(n) + ------ F(n) F(n + 1) 12122 1102 1102 6061 14679 8 65641 8 2 195185 9 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 24244 418 12122 2199 3 6 1974 3 7 8871 4 5 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 551 551 6061 16684 4 6 183270 5 2832 6 3 - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 6061 6061 6061 8221 6 4 639 7 2 472 2 8 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + --- F(n) F(n + 1) 551 12122 319 2733 5 51 6 + ----- F(n + 1) - --- F(n + 1) 24244 551 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 627 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 31 4 17 5 42 2 3 G(n) = --- F(n) F(n + 1) + -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 11 27 2 4 123 3 2 19 3 3 - -- F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) F(n + 1) 11 22 11 24 4 16 4 2 29 5 + -- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 11 11 11 2 13 2 5 + 5/22 F(n + 1) - 5/22 F(n + 1) - -- F(n) - 9/22 F(n) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 628 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(n + j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 2 4 3 G(n) = 1/60 F(n) (143 F(n + 1) + 1245 F(n) F(n + 1) + 90 F(n) F(n + 1) 2 3 5 - 108 F(n + 1) + 50 F(n) F(n + 1) - 820 F(n) F(n + 1) 2 2 4 5 6 - 115 F(n) F(n + 1) - 25 F(n + 1) - 1079 F(n) F(n + 1) + 155 F(n) 4 - 35 F(n) + 499 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 629 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(n + j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 293 2099 18960 3 5 22623 2 2 G(n) = --- F(n) - ---- F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 638 29 319 23860 7 1735 2 6 119 2 3 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 29 29 22 227 3 2 133 4 19225 8 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + ----- F(n + 1) 22 22 58 9170 3 2207 48484 3 71 5 + ---- F(n) F(n + 1) - ---- + ----- F(n) F(n + 1) + -- F(n + 1) 319 638 319 22 104614 4 - ------ F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 630 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(n + j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 85 131 6 G(n) = 5/22 F(n + 1) + 8/11 F(n) - -- F(n) F(n + 1) - --- F(n) F(n + 1) 11 11 46 2 14 2 299 2 5 - -- F(n) F(n + 1) + -- F(n) + --- F(n) F(n + 1) 11 11 11 3 3 128 3 4 388 4 3 - 31/2 F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 11 11 3 5 2 4 - 5/22 F(n + 1) + 6 F(n) F(n + 1) - 23/2 F(n) F(n + 1) 4 2 303 2 + 51/2 F(n) F(n + 1) + --- F(n) F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 631 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 2 2 6 G(n) = -1/12 F(n) (405 F(n) F(n + 1) - 10 F(n) F(n + 1) + 289 F(n) F(n + 1) 6 3 5 7 - 222 F(n) F(n + 1) + 129 F(n + 1) + 4 F(n + 1) - 135 F(n + 1) 4 2 3 7 - 54 F(n) F(n + 1) + 124 F(n) F(n + 1) + 44 F(n) + 50 F(n) 4 4 3 5 - 46 F(n) F(n + 1) - 520 F(n) F(n + 1) - 58 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 632 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 260 2 6 179 3 760 5 3 G(n) = --- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 11 22 11 45 2 144 2 287 2 2 - -- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 22 11 11 470 3 5 3 37 3 12 4 - --- F(n) F(n + 1) + 5/22 F(n + 1) - -- F(n) - -- F(n) 11 22 11 3 4 19 3 2 5 + 15/2 F(n) F(n + 1) + -- F(n) F(n + 1) - 5/22 - 10 F(n) F(n + 1) 11 5 2 4 3 325 4 4 - 95/2 F(n) F(n + 1) + 75/2 F(n) F(n + 1) - --- F(n) F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 633 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 23 5 312 3206 5 185 G(n) = --- F(n + 1) + ---- F(n) + ---- F(n + 1) + 2/5 F(n) + --- 10 1595 1595 638 25 5 3 4775 4 4 4 + -- F(n) F(n + 1) + ---- F(n) F(n + 1) - 9/2 F(n) F(n + 1) 29 319 3 5 430 7 4 - 10 F(n) F(n + 1) + --- F(n) F(n + 1) + 9/2 F(n) F(n + 1) 319 394 3 7155 7 15365 6 2 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 638 3117 4 - ---- F(n) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 634 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(n + 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 2 2 G(n) = -1/12 F(n) (74 F(n) F(n + 1) - 158 F(n) F(n + 1) - 320 F(n) F(n + 1) 5 3 2 - 36 F(n + 1) + 44 F(n + 1) - 2815 F(n + 1) + 1095 F(n) F(n + 1) 2 5 3 4 2 3 - 420 F(n) F(n + 1) + 6285 F(n) F(n + 1) + 88 F(n) F(n + 1) 6 7 3 - 6690 F(n) F(n + 1) + 2805 F(n + 1) - 6 F(n) + 54 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 635 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(n + 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 386 2 221535 9 185 9 3 3 G(n) = ---- F(n) + ------ F(n) + --- F(n + 1) + 12 F(n) F(n + 1) 319 6061 638 468700 3 2 542430 4 2166 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 6061 6061 319 636345 2 7 942255 3 6 3967 4 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 12122 638 123008 2 3 415995 8 4 2 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - 22 F(n) F(n + 1) 6061 12122 2 4 5 185 2 177835 + 11 F(n) F(n + 1) - 13/2 F(n) F(n + 1) - --- F(n + 1) - ------ F(n) 638 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 636 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 6 9 745 G(n) = -135/2 F(n + 1) + 955/6 F(n) F(n + 1) + 2275 F(n + 1) - --- F(n + 1) 12 85 13 3 3 26555 5 + -- F(n) - -- F(n) + 135/2 F(n + 1) - ----- F(n + 1) 12 12 12 11413 4 8 2 + ----- F(n) F(n + 1) - 22525/4 F(n) F(n + 1) + 27/4 F(n) F(n + 1) 12 2 3 2 5 3 6 - 3113/6 F(n) F(n + 1) + 40/3 F(n) F(n + 1) + 18125/4 F(n) F(n + 1) 301 2 2 7 3 2 - --- F(n) F(n + 1) + 1425/4 F(n) F(n + 1) + 611/2 F(n) F(n + 1) 12 3 4 - 925/6 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 637 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 95895 677201 615 8 G(n) = ------- F(n) - ------ F(n + 1) + --- F(n) F(n + 1) 315172 630344 572 2081 4 3710296 5 369 2 3 - ---- F(n) F(n + 1) - ------- F(n) F(n + 1) - --- F(n) F(n + 1) 572 1242505 572 2337 2 7 3528208 9 8519392 10 - ---- F(n) F(n + 1) + ------- F(n) F(n + 1) - ------- F(n + 1) 1144 248501 1242505 118443 10 615 9 369 7 2 - ------ F(n) + --- F(n) + --- F(n) F(n + 1) 12122 572 286 109205908 6 4 118251053 9 - --------- F(n) F(n + 1) + --------- F(n) F(n + 1) 1242505 2485010 212064199 8 2 3075 8 2000534 - --------- F(n) F(n + 1) + ---- F(n) F(n + 1) - ------- F(n) F(n + 1) 2485010 1144 1242505 156946186 7 3 163708 2 11410494 6 + --------- F(n) F(n + 1) - ------ F(n + 1) + -------- F(n + 1) 1242505 65395 1242505 1431 5 + ---- F(n + 1) 1144 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 638 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(n + 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 651 3 3 9675 2 7 111 2 4 G(n) = --- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 11 29 11 201525 8 364829 4 117523 2 3 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 58 638 319 2601 82108 3 2 155025 3 6 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 319 319 58 1231 5 80575 9 1033 2 7763 2 - ---- F(n) F(n + 1) + ----- F(n + 1) - ---- F(n) - ---- F(n + 1) 22 58 638 319 36963 4861 424661 5 267 6 - ----- F(n + 1) + ---- F(n) - ------ F(n + 1) + --- F(n + 1) 638 638 319 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 639 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(n + 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3345839 6 3534 2 299521 2 76745 3 G(n) = -------- F(n + 1) - ---- F(n) - ------ F(n + 1) - ----- F(n + 1) 638 319 638 638 663 3 165700 10 2645 7 30121 2 + --- F(n) + ------ F(n + 1) + ---- F(n + 1) + ----- F(n) F(n + 1) 319 29 22 638 94625 2 8 535402 3 3 416775 9 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 58 319 29 1093057 2 4 175 2 5 2935 3 4 - ------- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 11 11 312700 3 7 31409 607980 5 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 29 319 319 6315 6 4537 2 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 22 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 640 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3677 4 65 2 6 3980 3 5 G(n) = - 9/11 - ---- F(n + 1) - -- F(n) F(n + 1) + ---- F(n) F(n + 1) 22 11 11 3 4 6 86 3 - 325 F(n) F(n + 1) + 345 F(n) F(n + 1) + -- F(n) F(n + 1) 11 1255 2 1845 8 3195 3 35 3 - ---- F(n) F(n + 1) + ---- F(n + 1) + ---- F(n + 1) - -- F(n) 22 11 22 11 7 4450 7 387 2 - 145 F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 11 22 288 2 2 2 5 766 3 - --- F(n) F(n + 1) + 20 F(n) F(n + 1) + --- F(n) F(n + 1) 11 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 641 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(2 n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 6 G(n) = -1/12 F(n) (-87 + 337 F(n + 1) + 15 F(n) - 330 F(n + 1) 5 2 6 3 3 + 750 F(n) F(n + 1) + 225 F(n) F(n + 1) - 870 F(n) F(n + 1) 2 2 4 8 - 2176 F(n) F(n + 1) - 12897 F(n + 1) + 12975 F(n + 1) 3 7 - 97 F(n) F(n + 1) + 5469 F(n) F(n + 1) - 31575 F(n) F(n + 1) 2 4 3 3 5 + 210 F(n) F(n + 1) + 751 F(n) F(n + 1) + 27300 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 642 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(2 n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3745 8 17 3185 7 G(n) = ----- F(n) F(n + 1) + -- F(n) + ---- F(n) F(n + 1) 11 11 11 669 3 1059 3 2 18005 2 7 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 22 22 22 15075 2 6 2397 2 2 765 3 5 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 22 22 11 14745 4 4 420 3 6 158 4 + ----- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 22 11 11 7521 4 1501 2 3 49 4 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + -- F(n) 22 11 11 3216 3 9 4 5 - ---- F(n) F(n + 1) - 5/22 F(n + 1) - 1505/2 F(n) F(n + 1) 11 8 + 5/22 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 643 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 185 3 7213 2 309 2 G(n) = ---- F(n + 1) - ---- F(n) + --- F(n) F(n + 1) 638 638 638 1964400 4 2 1471 3 4 3 - ------- F(n) F(n + 1) + ---- F(n) - 55/2 F(n) F(n + 1) 319 638 3 4 32390 9 2343045 2 8 - 35/2 F(n) F(n + 1) - ----- F(n) F(n + 1) + ------- F(n) F(n + 1) 11 319 2 5 2835743 2 4 205515 3 7 + 10 F(n) F(n + 1) + ------- F(n) F(n + 1) - ------ F(n) F(n + 1) 638 319 3645215 4 6 907356 5 5 2 - ------- F(n) F(n + 1) - ------ F(n) F(n + 1) + 45 F(n) F(n + 1) 638 319 1143382 3 3 938852 8871 2 + ------- F(n) F(n + 1) + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 638 185 10 + --- F(n + 1) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 644 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(2 n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 6 5 4 5 G(n) = -1/240 F(n) (20460 F(n) F(n + 1) + 2740 F(n) - 21320 F(n) F(n + 1) 5 2 5 4 6 - 4936 F(n) F(n + 1) - 3070 F(n) F(n + 1) + 2500 F(n) F(n + 1) 6 3 7 2 5 - 61955 F(n) F(n + 1) + 81910 F(n) F(n + 1) + 2170 F(n + 1) 7 9 6 + 2932 F(n + 1) - 1055 F(n + 1) - 6100 F(n) F(n + 1) - 1355 F(n + 1) 4 7 4 3 - 19105 F(n) F(n + 1) - 580 F(n) + 9560 F(n) F(n + 1) 2 3 - 64 F(n) F(n + 1) - 2732 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 645 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 531 3655 4 2 6 4 G(n) = --- F(n + 1) - ---- F(n) F(n + 1) - 2785/4 F(n) F(n + 1) 176 44 102675 6 3 1799 5 5 9015 5 4 + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 176 11 44 228 5 6150 4 6 4 5 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) - 375 F(n) F(n + 1) 11 11 8397 4 5065 3 7 13925 3 6 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 176 22 88 18531 3 2 135 5 129 2 295 9 - ----- F(n) F(n + 1) - --- F(n) - --- F(n + 1) + --- F(n + 1) 88 22 44 176 93 6 6079 3 3 393 5 63 6 + -- F(n + 1) + ---- F(n) F(n + 1) - --- F(n + 1) - -- F(n) 22 22 88 22 67 10 - -- F(n + 1) 44 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 646 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(3 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 4 2 6 G(n) = -1/6 F(n) (93 F(n + 1) + 64 F(n) F(n + 1) - 100 F(n) F(n + 1) 3 8 4 + 121 F(n) F(n + 1) + 2750 F(n + 1) - 31 F(n) F(n + 1) - 2749 F(n + 1) 3 5 3 7 + 5900 F(n) F(n + 1) + 1148 F(n) F(n + 1) - 6650 F(n) F(n + 1) 3 3 2 5 - 234 F(n) F(n + 1) - 10 + 4 F(n) + 202 F(n) F(n + 1) 2 2 6 - 420 F(n) F(n + 1) - 88 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 647 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 105 10 1182 9 1362 5 2061 4 5 G(n) = --- F(n + 1) + ---- F(n) - ---- F(n) + ---- F(n) F(n + 1) 638 319 319 319 189 2 7 567 4 9375 6 3 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 638 638 14395 6 4 180 10 399 3 2 + ----- F(n) F(n + 1) + --- F(n) + --- F(n) F(n + 1) 638 319 319 313 2 8 304 819 7 2 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 319 4 6 487 4 2 837 3 3 + 8/29 F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 58 319 5805 8 9806 7 3 699 9 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 319 1950 8 2 105 9 + ---- F(n) F(n + 1) - --- F(n + 1) 319 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 648 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 10 7 4 2 G(n) = 5/22 F(n + 1) - 5/22 F(n + 1) - 91/4 F(n) F(n + 1) 217 3 7 14 6 5 - --- F(n) F(n + 1) + -- F(n) F(n + 1) + 3/22 F(n) F(n + 1) 22 55 27 63 2 91 6 - -- F(n) F(n + 1) - -- F(n) F(n + 1) + --- F(n) F(n + 1) 22 55 110 11067 5 5 1186 5 2485 4 6 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 110 55 44 103 4 3 57 2 17 7 3317 6 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + -- F(n) - ---- F(n) 22 110 22 220 3367 10 43 5 2 633 6 4 + ---- F(n) + -- F(n) F(n + 1) + --- F(n) F(n + 1) 220 22 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 649 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 12769 6 3 21671 9 G(n) = ----- F(n + 1) + 16/3 + 51/2 F(n) F(n + 1) + ----- F(n) F(n + 1) 410 615 26893 7 3 7 85093 6 4 + ----- F(n) F(n + 1) - 163/6 F(n) F(n + 1) - ----- F(n) F(n + 1) 205 410 2 6 2677 5 7 + 181/6 F(n) F(n + 1) + ---- F(n) F(n + 1) - 175/6 F(n) F(n + 1) 205 2 2 7678 9 2577 + 5/3 F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 123 205 6331 10 8 10 107 2 - ---- F(n + 1) - 25/6 F(n) - 19/6 F(n) - --- F(n + 1) 205 410 8 4 6337 8 2 + 20/3 F(n + 1) - 12 F(n + 1) - ---- F(n) F(n + 1) 410 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 650 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 4 2 5 225 2 4 G(n) = 72 F(n) F(n + 1) - 12 F(n) F(n + 1) + --- F(n) F(n + 1) 22 2 6 86 13 3 - 9/11 F(n) F(n + 1) - 66 F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) 11 22 35 2 555 6 7 195 2 + -- F(n) - --- F(n + 1) + 29 F(n + 1) + --- F(n) F(n + 1) 22 22 22 690 3 3 630 5 2 633 3 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + 25 F(n + 1) - --- F(n + 1) 11 11 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 651 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 3 3 4 G(n) = -1/12 F(n) (-40 F(n + 1) - 2622 F(n + 1) + 5970 F(n) F(n + 1) 2 3 2 6 + 70 F(n) F(n + 1) - 256 F(n) F(n + 1) - 6180 F(n) F(n + 1) 2 7 3 2 + 978 F(n) F(n + 1) + 2610 F(n + 1) - 145 F(n) F(n + 1) 2 5 4 3 - 540 F(n) F(n + 1) + 85 F(n) F(n + 1) + 46 F(n + 1) + 32 F(n) - 8 F(n)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 652 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 395 5 26245 7 2 12520 8 G(n) = --- F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 451 451 451 5259 8 7155 2 4 2532 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 902 638 319 15975 6 29599 52730 849 2 + ----- F(n + 1) + ----- F(n) - ----- F(n + 1) - --- F(n) 638 13079 13079 638 8040 2 1865 9 2995 9 7529 6 3 - ---- F(n + 1) + ---- F(n) + ---- F(n + 1) - ---- F(n) F(n + 1) 319 902 902 451 5350 4 19710 3 3 18360 5 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 451 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 653 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 46 2 5 329 3 4 4 4 30 7 G(n) = -- F(n) F(n + 1) - --- F(n) F(n + 1) - 20 F(n) F(n + 1) - -- F(n) 11 22 11 23 4 13 7 125 8 3 + -- F(n) - -- F(n + 1) - --- - 3/16 F(n + 1) + 15/8 F(n) F(n + 1) 16 22 176 3 5 255 6 2 257 6 + 95/8 F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 16 22 5 3 443 4 3 447 5 2 + 91/4 F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 22 22 3 4 + 4/11 F(n + 1) + 9/8 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 654 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 G(n) = 1/24 F(n) (187 F(n + 1) + 338 F(n) F(n + 1) - 146 F(n) F(n + 1) 8 2 4 2 6 6 - 2292 F(n + 1) - 410 F(n) F(n + 1) + 1795 F(n) F(n + 1) - 177 F(n) 4 2 6 5 3 + 2495 F(n + 1) - 173 + 147 F(n) - 205 F(n + 1) - 8374 F(n) F(n + 1) 5 7 2 2 + 574 F(n) F(n + 1) + 3840 F(n) F(n + 1) + 2126 F(n) F(n + 1) 8 + 275 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 655 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7320 4 3 2 412 3 8 G(n) = ---- F(n + 1) + 19 F(n) F(n + 1) - --- F(n) F(n + 1) - 670 F(n + 1) 11 11 95 4 9 2 7 + -- + 297 F(n) F(n + 1) + 670 F(n + 1) + 15 F(n) F(n + 1) 22 2 3 2 2 8 - 106 F(n) F(n + 1) + 223/2 F(n) F(n + 1) - 1630 F(n) F(n + 1) 7 3110 3 3 5 + 1630 F(n) F(n + 1) - ---- F(n) F(n + 1) - 1410 F(n) F(n + 1) 11 3 6 2 6 29 49 + 1410 F(n) F(n + 1) - 15 F(n) F(n + 1) - -- F(n) + -- F(n + 1) 22 22 5 - 672 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 656 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 4 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 180 4 153500 2 9 1203625 11 G(n) = ---- F(n + 1) - ------ F(n) F(n + 1) - ------- F(n + 1) 551 551 1102 575830 3 1290 3 1173 3 12089595 7 + ------ F(n + 1) - ---- F(n) - ---- F(n) F(n + 1) + -------- F(n + 1) 6061 6061 1102 12122 1290 1892425 2 5 22818 2 + ---- + ------- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 6061 6061 99 2 2 3016125 10 3918955 3 4 + ---- F(n) F(n + 1) + ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 1102 1102 12122 2300625 3 8 134983 2 279 3 - ------- F(n) F(n + 1) - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 1102 6061 1102 2061870 6 - ------- F(n) F(n + 1) 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 657 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 4 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 52375 11 3 47815 7 13 3 G(n) = ----- F(n + 1) - 1140/7 F(n + 1) - ----- F(n + 1) + -- F(n) 28 28 21 13375 2 9 327 3 2 557 2 + ----- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 28 28 84 4 6985 6 10 + 57/7 F(n) F(n + 1) + ---- F(n) F(n + 1) - 9375/2 F(n) F(n + 1) 12 100125 3 8 1089 2 99 2 3 + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) + -- F(n) F(n + 1) 28 28 28 6415 2 5 3 4 111 13 - ---- F(n) F(n + 1) + 3325/6 F(n) F(n + 1) + --- F(n + 1) - -- F(n) 12 28 21 111 5 - --- F(n + 1) 28 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 658 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 4 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 105 11 6075 2 5 153575 4 3 G(n) = --- F(n + 1) + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) 638 11 638 29615 6 2220605 2 9 1911 5 - ----- F(n) F(n + 1) - ------- F(n) F(n + 1) + ---- F(n) F(n + 1) 22 638 319 1769355 4 7 105 10 723 2 723 3 + ------- F(n) F(n + 1) - --- F(n + 1) + --- F(n) - --- F(n) 638 638 638 638 105 3 7 78480 3 8 105 2 8 - --- F(n) F(n + 1) + ----- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 638 93 3 3 216145 3 4 8763 2 4 - -- F(n) F(n + 1) - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 29 638 638 3381 36647 2 105 9 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + --- F(n) F(n + 1) 638 638 319 447395 10 3252 2 4533 4 2 + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 319 105 4 6 - --- F(n) F(n + 1) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 659 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 105 4 1973 2 2 2 G(n) = - --- + 6/11 F(n) + ---- F(n) F(n + 1) - 9/22 F(n) F(n + 1) 638 638 3 4025 3 4 629 2 - 6/11 F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 638 58 3375 6 7400 2 5 17525 4 3 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 638 243 3 105 3 - --- F(n) + --- F(n + 1) 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 660 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 4 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 5 2 G(n) = -1/84 F(n) (1375 F(n + 1) - 3125 F(n) F(n + 1) - 58 F(n) 3 3 2 4 3 + 3625 F(n) F(n + 1) - 875 F(n) F(n + 1) - 227 F(n) F(n + 1) 4 3 + 30 F(n + 1) + 417 F(n) F(n + 1) - 60 F(n) F(n + 1) 2 2 4 2 + 257 F(n) F(n + 1) + 58 F(n) - 1417 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 661 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 4 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 201 3 3 92203 3 4 42469 2 5 G(n) = --- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 12122 12122 4597 6 31835 2 5865 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 12122 6061 12122 111139 5 2 690 2 12753 2 - ------ F(n) F(n + 1) - ---- F(n + 1) + ----- F(n) 6061 6061 12122 2097 4 2 139269 4 3 2187 5 + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 638 12122 638 822 2 4 12753 3 690 3 - --- F(n) F(n + 1) - ----- F(n) + ---- F(n + 1) 319 12122 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 662 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 4 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2540 2 6 15 2 3 1134 2 2 G(n) = ---- F(n) F(n + 1) - -- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 11 319 6077 6 2 51 4 3 2 + ---- F(n) F(n + 1) + -- F(n) F(n + 1) - 3/2 F(n) F(n + 1) 638 22 831 3 1080 7 437 7 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 319 3247 3 5 105 36 8 4 177 - ---- F(n) F(n + 1) + --- + --- F(n) + 9/22 F(n) F(n + 1) - --- F(n) 319 638 319 638 105 - --- F(n + 1) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 663 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 265 3 6625 3 5 46075 7 G(n) = --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 21 14 84 1067 2 2 2 4 317 - ---- F(n) F(n + 1) + 8 F(n) F(n + 1) - --- F(n) F(n + 1) 28 42 18811 4 277 2 19 2 2685 3 - ----- F(n + 1) + --- F(n + 1) + -- F(n) + ---- F(n) F(n + 1) 84 12 14 28 277 6 2 6 3 3 - --- F(n + 1) + 25/4 F(n) F(n + 1) - 169/3 F(n) F(n + 1) 12 18925 8 19 5 + ----- F(n + 1) - -- + 319/6 F(n) F(n + 1) 84 14 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 664 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 4 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2265 4 3 4365 3 4 690 3 G(n) = ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n + 1) 319 319 6061 8940 3 750 4 690 5805 2 5 - ---- F(n) + --- F(n) + ---- - ---- F(n) F(n + 1) 6061 551 6061 638 98375 5 3 12825 5 2 8199 2 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 1102 638 12122 45492 2 17853 2 2 32525 2 6 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 551 1102 10187 3 37900 3 5 881 3 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 1102 551 1102 30575 4 4 + ----- F(n) F(n + 1) 1102 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 665 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 54595 8 8185 7 2 208 5 127 5 G(n) = ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n) - ---- F(n + 1) 1276 319 11 1276 5822 9 337 189 2 42 6 210 6 + ---- F(n) + ---- F(n + 1) - --- F(n + 1) + --- F(n + 1) + --- F(n) 319 1276 638 319 319 39 4 2 70 6 3 1689 5 + -- F(n) F(n + 1) - -- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 11 638 3205 2 7 2553 4 5 18 2 4 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - -- F(n) F(n + 1) 1276 638 11 579 3 3 4867 3 2 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 666 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 4 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 8 3 3 G(n) = 1/420 F(n) (-3745 F(n) + 860 F(n) + 10325 F(n) F(n + 1) 5 3 2 2 2 - 28980 F(n) F(n + 1) + 2885 F(n) + 623 F(n) F(n + 1) 6 2 7 2 + 12792 F(n) F(n + 1) - 5375 F(n) F(n + 1) - 310 F(n + 1) 3 7 - 1755 F(n) F(n + 1) + 8799 F(n) F(n + 1) - 10279 F(n) F(n + 1) 2 4 8 5 - 10325 F(n) F(n + 1) + 370 F(n + 1) + 2065 F(n) F(n + 1) 2 6 + 22050 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 667 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 4 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 555 2 17495 5 5 167553 5 G(n) = --- F(n) F(n + 1) - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 319 638 12760 597 2 5 3789 9 17977 5 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 2552 1276 63895 8 2 96675 7 3 50189 6 4 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 1276 2552 1276 15969 6 597 7 597 6 3177 5 2 + ----- F(n + 1) - --- F(n) + --- F(n) - ---- F(n) F(n + 1) 3190 638 638 319 1686 6 2623 21 3 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + -- F(n + 1) 319 1160 29 138 6 702 2 567 7 + --- F(n) F(n + 1) - --- F(n + 1) - --- F(n + 1) 319 145 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 668 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 6 2 G(n) = 1/420 F(n) (-2195 F(n) F(n + 1) - 100 - 160 F(n + 1) + 480 F(n + 1) 6 8 5 3 4 6 + 400 F(n) + 120 F(n) - 240 F(n) F(n + 1) + 9550 F(n) F(n + 1) 3 2 8 2 6 - 33 F(n) F(n + 1) - 1100 F(n) F(n + 1) + 24 F(n) F(n + 1) 2 4 2 2 8 2 + 3000 F(n) F(n + 1) - 459 F(n) F(n + 1) + 9675 F(n) F(n + 1) 7 3 7 6 4 - 14725 F(n) F(n + 1) + 288 F(n) F(n + 1) + 5450 F(n) F(n + 1) 5 5 - 9975 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 669 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 3 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 497 2 20 10 20 11 59 2 G(n) = --- F(n) F(n + 1) + --- F(n + 1) - --- F(n + 1) - --- F(n) 638 319 319 319 86 11 10 4047 10 461 9 2 - --- F(n) - 6/11 F(n) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 638 1595 1664 9 25780 8 3 2612 8 2 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 21717 7 4 588 5 5 90617 5 2 - ----- F(n) F(n + 1) + --- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 3190 2113 5 169 4 7 2558 4 6 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 110 319 7057 4 3 684 4 2 804 3 8 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 1595 319 319 1556 3 7 25 2 41 - ---- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 319 22 58 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 670 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 4 4 G(n) = 1/12 F(n) (284 F(n + 1) + 11 F(n) + F(n) - 3 F(n + 1) 3 2 2 - 84 F(n) F(n + 1) + 6 F(n) F(n + 1) - 25 F(n) F(n + 1) 2 4 3 6 + 175 F(n) F(n + 1) + 10 F(n) F(n + 1) - 275 F(n + 1) 5 3 3 + 625 F(n) F(n + 1) - 725 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 671 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 3 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4492 2 61 5 48 2 4 G(n) = ----- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 319 22 11 145 4 2 20 2 38 2 20 3 - --- F(n) F(n + 1) - --- F(n + 1) - --- F(n) + --- F(n + 1) 22 319 319 319 281 3 13 3 3 2625 3 4 - --- F(n) + -- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 11 319 11900 4 3 716 9275 6 + ----- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 638 1288 2 19775 2 5 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 672 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 15465 3 334041 2 690 7 690 4 G(n) = ------ F(n) - ------ F(n) F(n + 1) - ---- F(n + 1) + ---- F(n + 1) 6061 6061 6061 6061 2718 4 873229 3 2055 7 - ---- F(n) + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 12122 29 97953 2 304353 2 2 4830 2 6 + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 6061 12122 29 209280 3 4 4950 4 4 1482795 2 5 - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) - ------- F(n) F(n + 1) 6061 29 12122 859785 4 3 655335 6 730 3 5 + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) + --- F(n) F(n + 1) 6061 12122 29 35063 3 + ----- F(n) F(n + 1) 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 673 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 7 2 G(n) = -1/12 F(n) (-1135 F(n + 1) + 1125 F(n + 1) + 415 F(n) F(n + 1) 2 2 3 3 4 - 108 F(n) F(n + 1) + 48 F(n) F(n + 1) + 2625 F(n) F(n + 1) 4 2 5 + 4 F(n + 1) - 2 F(n) + 12 F(n) F(n + 1) - 300 F(n) F(n + 1) 3 2 3 6 - 48 F(n) F(n + 1) + 14 F(n) - 2650 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 674 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 68500 2 6 3119 3 10773 2 2 G(n) = ------ F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 319 29275 7 2317 2 29021 3 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 638 319 336 4 3 134275 4 4 141 3 20 3 - --- F(n) F(n + 1) + ------ F(n) F(n + 1) + --- F(n) - --- F(n + 1) 11 638 319 319 20 477 4 289 6 57 3 4 + --- + --- F(n) - --- F(n) F(n + 1) + -- F(n) F(n + 1) 319 319 22 22 631 2 5 7907 2 6000 3 5 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 22 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 675 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 241 4 697 4 5 6 G(n) = --- F(n) F(n + 1) - --- F(n) F(n + 1) + 9/4 F(n) F(n + 1) 36 36 7567 6 3 2120 7 2 14755 8 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 108 27 108 2 5 6487 5 175 5 61 3 + 55/6 F(n) F(n + 1) + ---- F(n) - --- F(n + 1) - -- F(n + 1) 108 54 60 19 8 83 6 67 2 + -- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 27 12 15 329 5 2 61 7 7 3131 9 - --- F(n) F(n + 1) + -- F(n + 1) - 1/12 F(n) - ---- F(n) 30 60 54 175 + --- F(n + 1) 54 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 676 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 35527 9 2725 8 3681 7 G(n) = ----- F(n) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 2552 58 638 3411 6 2 50305 7 2 373 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - --- F(n) 290 1276 232 1941 5 3 4706 5 20 93745 5 4 + ---- F(n) F(n + 1) - ---- F(n) - --- - ----- F(n) F(n + 1) 638 319 319 2552 2264 7 20 9 706 4 4 - ---- F(n) F(n + 1) + --- F(n + 1) - --- F(n) F(n + 1) 1595 319 319 992 4 5 4 975 8 + --- F(n) F(n + 1) + 11/4 F(n) F(n + 1) - ---- F(n) F(n + 1) 29 2552 5691 2 2 37 3 159 8 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) 3190 110 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 677 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 8 3 G(n) = -3989 F(n + 1) - 160 F(n + 1) - 17/4 F(n) F(n + 1) 2 6 2 2 10 + 10 F(n) F(n + 1) - 608/3 F(n + 1) - 13/4 F(n) + 12575/3 F(n + 1) 4 431 9791 3 3 + 639/4 F(n + 1) + --- F(n) F(n + 1) + 1/4 + ---- F(n) F(n + 1) 12 12 2 8 5 3 + 2525/3 F(n) F(n + 1) + 6439/4 F(n) F(n + 1) - 65 F(n) F(n + 1) 12761 2 4 2 2 9 - ----- F(n) F(n + 1) + 21 F(n) F(n + 1) - 31300/3 F(n) F(n + 1) 12 7 3 7 3 5 + 385 F(n) F(n + 1) + 8200 F(n) F(n + 1) - 350 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 678 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 2 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2415 8 59931 2 1733 3 G(n) = ----- F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 418 94 209 178740 3 4 152935 3 201500 11 + ------ F(n) F(n + 1) - ------ F(n + 1) + ------ F(n + 1) 47 94 517 5063 3 1275 8 28575 11 58160 7 + ---- F(n) - ---- F(n) - ----- F(n) + ----- F(n + 1) 94 418 517 47 909 3 7135 2 6 96875 2 9 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 418 418 94 160050 6 3945 7 119025 10 - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 47 209 94 27995 2 5 3603 2 2 9374 2 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 47 418 47 34 4 514 + -- F(n + 1) + --- 11 209 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 679 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 16 7 3 2 105 6 G(n) = --- F(n) - 2/5 F(n + 1) + 9/110 F(n + 1) + --- F(n) F(n + 1) 11 22 3 3 4 2 157 4 3 + 1/11 F(n) F(n + 1) - 4 F(n) F(n + 1) + --- F(n) F(n + 1) 11 5 2 113 2 60 3 4 - 46/5 F(n) F(n + 1) - --- F(n) F(n + 1) - -- F(n) F(n + 1) 110 11 71 5 14 69 7 17 6 + -- F(n) F(n + 1) + -- F(n) F(n + 1) + --- F(n + 1) - -- F(n + 1) 55 55 110 55 6 + 5/11 F(n) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 680 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 475 4 1751 3 20 8 515 3 G(n) = --- F(n) - ---- F(n) + --- F(n + 1) + --- F(n) F(n + 1) 638 638 319 638 38065 2 11840 3 4 46235 4 4 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 319 319 35909 3 17585 7 11137 2 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 638 319 638 39855 2 6 450 2 2 11205 3 5 + ----- F(n) F(n + 1) - --- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 29 319 42180 2 5 18640 6 48980 4 3 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 319 20 7 - --- F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 681 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 9 5 4875 3 4 G(n) = 2180 F(n + 1) - 8537/4 F(n + 1) - ---- F(n) F(n + 1) 28 5659 2 3 745 2 8 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) - 10765/2 F(n) F(n + 1) 12 84 5175 6 3 6 2981 3 2 + ---- F(n) F(n + 1) + 8755/2 F(n) F(n + 1) + ---- F(n) F(n + 1) 28 12 2545 2 4 2 7 - ---- F(n) F(n + 1) + 2783/3 F(n) F(n + 1) + 595/2 F(n) F(n + 1) 84 2 5 55 3 181 + 75/7 F(n) F(n + 1) - -- F(n) + --- F(n) - 183/4 F(n + 1) 42 42 2175 3 2175 7 + ---- F(n + 1) - ---- F(n + 1) 28 28 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 682 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 75 2 6 12 4 3 179 3 G(n) = -- F(n) F(n + 1) + 5/22 + -- F(n) - 7/22 F(n) - --- F(n) F(n + 1) 22 11 22 2 5 3 5 289 3 4 - 13/2 F(n) F(n + 1) + 75 F(n) F(n + 1) - --- F(n) F(n + 1) 44 2175 5 3 511 5 2 171 3 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 22 44 11 12 2 651 2 2 325 7 - -- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 11 22 22 157 2 157 6 7 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - 5/22 F(n + 1) 44 44 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 683 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 15447 4 927 4 14097 8 1007 8 961 G(n) = ----- F(n) + ---- F(n + 1) - ----- F(n) - ---- F(n + 1) + --- F(n) 319 1276 319 1276 638 20 5 26963 5 8345 2 2 567 2 3 + --- F(n + 1) + ----- F(n) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 319 638 1276 319 379 3 6 13791 4 17143 5 3 + --- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 22 319 638 1565 5 4 110665 6 2 6785 7 - ---- F(n) F(n + 1) + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 22 1276 58 3325 7 2 556 8 3631 3 5 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 22 11 319 9 - 45 F(n) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 684 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 9 181 9 211 5 G(n) = F(n) + 5/22 F(n + 1) + --- F(n) - 5/22 - 1/22 F(n) - --- F(n) 11 11 2 2 2 6 338 4 5 + 9/5 F(n) F(n + 1) - 19/5 F(n) F(n + 1) + --- F(n) F(n + 1) 11 19 7 38 2 3 27 2 7 + -- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 10 11 22 21 3 635 5 4 1009 7 2 + -- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 10 22 22 545 8 3 5 3 + --- F(n) F(n + 1) - F(n) F(n + 1) + 21/2 F(n) F(n + 1) 11 6 2 - 35/2 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 685 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 6 3 7 3 6 G(n) = 7775/2 F(n) F(n + 1) + 175 F(n) F(n + 1) + 970 F(n) F(n + 1) 8424 3 3 3 2 2 7 - ---- F(n) F(n + 1) + 22 F(n) F(n + 1) + 85/2 F(n) F(n + 1) 11 21613 2 4 2 3 9 + ----- F(n) F(n + 1) - 177/2 F(n) F(n + 1) + 1900 F(n) F(n + 1) 22 8 32471 5 4 - 2345/2 F(n) F(n + 1) - ----- F(n) F(n + 1) + 439/2 F(n) F(n + 1) 22 2128 2 45 2 9 4251 6 12 + ---- F(n + 1) + -- F(n) + 955/2 F(n + 1) - ---- F(n + 1) - -- F(n) 11 11 22 11 17 403 5 + -- F(n + 1) - --- F(n) F(n + 1) - 957/2 F(n + 1) 22 11 2 8 - 9325/2 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 686 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 4 6 8 G(n) = 1/120 F(n) (-995 F(n) + 25700 F(n) F(n + 1) - 133 F(n + 1) 7 6 2 8 2 - 1920 F(n) F(n + 1) + 775 - 4730 F(n) F(n + 1) + 45000 F(n) F(n + 1) 10 4 4 4 - 400 F(n + 1) - 782 F(n + 1) - 3535 F(n) F(n + 1) 7 3 9 3 - 31500 F(n) F(n + 1) + 7900 F(n) F(n + 1) + 2096 F(n) F(n + 1) 5 5 3 6 - 10800 F(n) F(n + 1) + 8524 F(n) F(n + 1) + 1600 F(n + 1) 2 10 5 5 - 1820 F(n) F(n + 1) - 980 F(n + 1) + 700 F(n) - 34700 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 687 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 5 537 2 G(n) = -5/22 F(n + 1) + 5/22 + 35 F(n) F(n + 1) - --- F(n) F(n + 1) 11 1200 2 5 331 2 2 158 2 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 11 22 11 7 525 6 1241 3 - 55 F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 22 4 4 2775 4 3 675 3 4 - 145 F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 22 22 27 3 2 6 4 26 3 + -- F(n) F(n + 1) + 125 F(n) F(n + 1) + 3/22 F(n) - -- F(n) 22 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 688 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 6 2 G(n) = 1/60 F(n) (119 F(n) F(n + 1) - 63 F(n + 1) + 285 + 28 F(n + 1) 6 8 3 3 3 5 - 65 F(n) + 20 F(n) + 630 F(n) F(n + 1) - 615 F(n) F(n + 1) 4 2 5 3 6 2 - 875 F(n) F(n + 1) - 7085 F(n) F(n + 1) + 10550 F(n) F(n + 1) 7 3 8 + 575 F(n) F(n + 1) - 3445 F(n) F(n + 1) + 220 F(n + 1) 4 + 161 F(n) F(n + 1) - 440 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 689 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 295467 6 705 3 2513469 2 11925 2 5 G(n) = ------ F(n + 1) + --- F(n) - ------- F(n) - ----- F(n) F(n + 1) 1111 319 32219 638 17725 2 8 15489 2 7050 10 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) 202 319 101 57121 2 4 5299759 95175 6 - ----- F(n) F(n + 1) + ------- F(n) F(n + 1) - ----- F(n) F(n + 1) 2222 64438 319 736246 5 4512 2 4850 9 - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 1111 319 101 178425 3 4 849523 3 3 27525 9 + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 638 2222 101 39990 3 17126481 2 79875 7 - ----- F(n + 1) - -------- F(n + 1) + ----- F(n + 1) 319 64438 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 690 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 15325 8 9 1209 2515 4 8869 4 G(n) = ------ F(n + 1) - 145 F(n + 1) - ---- + ---- F(n) + ---- F(n + 1) 22 11 22 11 5 11339 3 7 + 77/2 F(n + 1) - ----- F(n) F(n + 1) + 3375/2 F(n) F(n + 1) 22 32125 3 5 425 4 4 4 5 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) - 815 F(n) F(n + 1) 22 22 4 2 7 2085 3 + 505 F(n) F(n + 1) + 830 F(n) F(n + 1) + ---- F(n) F(n + 1) 11 3 2 3 6 4 - 375/2 F(n) F(n + 1) - 220 F(n) F(n + 1) - 217/2 F(n) F(n + 1) 1174 - 9/22 F(n) + ---- F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 691 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 5 3 5 G(n) = -1/12 F(n) (400 F(n) F(n + 1) - 26 F(n) - 53602 F(n + 1) 3 2 3 2 + 1785 F(n + 1) - 11100 F(n) F(n + 1) + 182 F(n) F(n + 1) 8 6 4 - 134250 F(n) F(n + 1) + 4200 F(n) F(n + 1) + 22964 F(n) F(n + 1) 2 9 7 - 665 F(n) F(n + 1) + 54600 F(n + 1) - 1775 F(n + 1) 3 6 3 4 3 2 + 111150 F(n) F(n + 1) - 4075 F(n) F(n + 1) + 5578 F(n) F(n + 1) 2 7 + 5550 F(n) F(n + 1) - 1014 F(n + 1) + 98 F(n)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 692 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3247 2 4331 9 9 4629 5 8 G(n) = ----- F(n) + ---- F(n) - 5/22 F(n + 1) - ---- F(n) - F(n) F(n + 1) 176 44 44 102 2 3 447 2 7 2385 5 4 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 55 55 44 35699 7 2 3383 10 10 - ----- F(n) F(n + 1) + ---- F(n) + 5/22 F(n + 1) 220 176 19 9 2157 2 4 1835 2 8 + -- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 22 88 176 2009 3 2 45645 5 31467 5 5 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 55 352 352 29341 7 3 5455 8 45951 8 2 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 352 22 176 17781 9 - ----- F(n) F(n + 1) 352 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 693 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(3 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = 1/6 F(n) (1 + 3 F(n + 1) - 3 F(n) F(n + 1) + 2 F(n) 2 2 3 4 + 15 F(n) F(n + 1) - 21 F(n) F(n + 1) + 3 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 694 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 19 3 90 2 4 223 2 G(n) = -- F(n) + -- F(n) F(n + 1) + 3/2 F(n) F(n + 1) - --- F(n + 1) 22 11 11 41 2 3 6 21 2 - -- F(n) - 5/22 F(n + 1) + 41/2 F(n + 1) + -- F(n) F(n + 1) 22 22 18 2 5 3 3 - -- F(n) F(n + 1) - 50 F(n) F(n + 1) + 42 F(n) F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 695 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 147 3 189 2 587 84127 6 2 G(n) = --- F(n) - --- F(n) F(n + 1) - ---- - ----- F(n) F(n + 1) 638 319 1914 3190 14755 7 1207 3 63 2 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 957 1595 638 43111 7 56351 2 2 8641 8 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n + 1) 4785 9570 1914 2798 8 105 3 3181 4 - ---- F(n) - --- F(n + 1) + ---- F(n + 1) 957 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 696 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 6 G(n) = -1/2 F(n) F(n + 1) + 22/3 F(n) F(n + 1) + 23 F(n + 1) 3 2 2 2 4 + 2/3 F(n) F(n + 1) - 1/3 F(n) F(n + 1) - 8 F(n) F(n + 1) + 1/6 2 2 4 3 3 - 7/6 F(n) - 23 F(n + 1) - 1/6 F(n + 1) + 55 F(n) F(n + 1) 5 - 53 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 697 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 65 3 897 2 7 21 4 G(n) = -- F(n) + --- F(n) F(n + 1) + 5/22 F(n + 1) - -- F(n) 22 22 22 4 467 6 2261 2 5 - 5/22 F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 22 971 4 3 15 3 343 2 - --- F(n) F(n + 1) + -- F(n) F(n + 1) - --- F(n) F(n + 1) 11 11 22 27 2 2 32 3 23 3 4 - -- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 11 11 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 698 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 2 3 G(n) = -44/3 F(n) F(n + 1) + 1/6 F(n) + 7/10 F(n) F(n + 1) + 1/10 F(n + 1) 3 4 7 449 5 2 + 41/6 F(n) F(n + 1) + 11/6 F(n) + --- F(n) F(n + 1) 30 6 7 - 59/6 F(n) F(n + 1) - 1/10 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 699 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 53 3 5 2 3 G(n) = 9 F(n + 1) - -- + 720 F(n) F(n + 1) - 19/2 F(n) F(n + 1) 11 2 2 7 1911 3 - 87 F(n) F(n + 1) - 930 F(n) F(n + 1) + ---- F(n) F(n + 1) 11 412 3 2 6 4042 4 + --- F(n) F(n + 1) + 85 F(n) F(n + 1) - ---- F(n + 1) 11 11 3 2 4 20 203 + 27 F(n) F(n + 1) - 37/2 F(n) F(n + 1) + -- F(n) - --- F(n + 1) 11 22 8 + 745/2 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 700 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 5 3 2 G(n) = -1/3 F(n) (-120 F(n) F(n + 1) + 9 F(n) - 70 F(n) F(n + 1) 6 7 3 - 1605 F(n) F(n + 1) + 675 F(n + 1) - 677 F(n + 1) 3 4 2 + 1530 F(n) F(n + 1) + 258 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 701 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 237 2 1127 3 8 G(n) = --- F(n) F(n + 1) + ---- F(n) F(n + 1) + 615/2 F(n + 1) 638 58 85761 3 117 2 35901 2 2 + ----- F(n) F(n + 1) - --- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 638 638 8295 7 13875 3 5 20 3 59 3 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) + --- F(n + 1) + --- F(n) 11 22 319 319 194831 4 280 2 6 697 - ------ F(n + 1) + --- F(n) F(n + 1) - --- 638 11 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 702 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 30329 5 9 2 62 G(n) = ------ F(n + 1) + 8855/6 F(n + 1) + 1/14 F(n) F(n + 1) + -- F(n) 21 21 1327 3 2419 3 2 2695 2 7 - ---- F(n + 1) + 1/21 F(n) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 42 14 12 53191 4 8 2 + ----- F(n) F(n + 1) - 14605/4 F(n) F(n + 1) - 1/14 F(n) F(n + 1) 84 27427 2 3 3 6 - ----- F(n) F(n + 1) + 17665/6 F(n) F(n + 1) 84 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 703 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 5 51 3 G(n) = 5/22 F(n + 1) + 7/2 F(n) F(n + 1) + -- F(n) F(n + 1) 22 3 4 3 6 + 293/2 F(n) F(n + 1) - 1/22 F(n) F(n + 1) - 172 F(n) F(n + 1) 100 2 32 2 2 333 2 16 3 - --- F(n) F(n + 1) - -- F(n) F(n + 1) + --- F(n) F(n + 1) + -- F(n) 11 11 11 11 7 773 3 - 5/11 + 141/2 F(n + 1) - --- F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 704 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 23 959 2 2 2517 8 G(n) = --- F(n) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 58 638 1276 2925 6 2 299 3 353 8 20 8 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) + --- F(n + 1) 638 638 638 319 217 4 2737 9 20 9 357 2 3 - --- F(n) + ---- F(n) - --- F(n + 1) + --- F(n) F(n + 1) 638 638 319 319 393 3 5 59 3 6 19647 4 + --- F(n) F(n + 1) - --- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 1276 2725 4 4 15577 4 5 96815 6 3 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 1276 1276 240 7 48385 7 2 + --- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 705 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 29 63 3 2 6687 4 G(n) = -- F(n) - -- F(n + 1) + 33/2 F(n) F(n + 1) - ---- F(n + 1) 22 11 22 549 3 73 2 6 1515 3 + --- F(n) F(n + 1) - -- + 45 F(n) F(n + 1) + ---- F(n) F(n + 1) 22 22 11 4 7 2 2 - 12 F(n) F(n + 1) - 760 F(n) F(n + 1) - 62 F(n) F(n + 1) 2 3 3 5 5 - 13/2 F(n) F(n + 1) + 615 F(n) F(n + 1) + 11/2 F(n + 1) 8 + 615/2 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 706 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 1313 5 39 2 2 8 G(n) = ---- F(n) - -- F(n) - 5/22 F(n + 1) - 715 F(n) F(n + 1) 11 22 4 2160 4 2 7 + 1659/2 F(n) F(n + 1) + ---- F(n) F(n + 1) + 3455/2 F(n) F(n + 1) 11 11475 2 3 5 4 5 - ----- F(n) F(n + 1) + 15 F(n) F(n + 1) - 2895/2 F(n) F(n + 1) 22 5 4 5 2521 - 145/2 F(n) F(n + 1) + 5 F(n) F(n + 1) - ---- F(n) + 5/22 F(n + 1) 22 4 2 105 2 4 - 5/2 F(n) F(n + 1) - --- F(n) F(n + 1) - 25/2 F(n) F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 707 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 5 G(n) = 1/84 F(n) (-4 + 8 F(n) + 308 F(n) - 48 F(n) F(n + 1) 4 2 4 4 5 3 - 120 F(n) F(n + 1) + 45885 F(n) F(n + 1) - 6440 F(n) F(n + 1) 3 7 2 2 - 18865 F(n) F(n + 1) + 18865 F(n) F(n + 1) + 9150 F(n) F(n + 1) 6 2 4 2 6 + 24 F(n) + 120 F(n) F(n + 1) - 46515 F(n) F(n + 1) 3 2 - 2416 F(n) F(n + 1) + 48 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 708 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6933857 53773547 5 G(n) = ------- F(n) F(n + 1) - -------- F(n) F(n + 1) 2485010 1242505 19745822 2 8 26646179 2 4 - -------- F(n) F(n + 1) - -------- F(n) F(n + 1) 248501 2485010 173110693 8 2 11886956 7 3 - --------- F(n) F(n + 1) + -------- F(n) F(n + 1) 1242505 42845 50469644 9 31087432 10 3207 3 22974967 2 - -------- F(n) F(n + 1) - -------- F(n) + ---- F(n) + -------- F(n) 1242505 1242505 6061 1242505 49096811 9 7551 2 11691 2 + -------- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 1242505 12122 12122 690 3 690 10 - ---- F(n + 1) + ---- F(n + 1) 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 709 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 29215 9 751 3 11015 2165 G(n) = ----- F(n + 1) - --- F(n) F(n + 1) - ----- F(n + 1) + ---- F(n) 22 319 319 638 347 4 97979 2 3 825245 5 - --- F(n + 1) - ----- F(n) F(n + 1) - ------ F(n + 1) 638 319 638 4955 2 7 28935 3 6 8 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - 6585/2 F(n) F(n + 1) 22 11 417 3 387 56330 3 2 585 2 2 + --- F(n) F(n + 1) + --- + ----- F(n) F(n + 1) + --- F(n) F(n + 1) 319 638 319 638 179953 4 + ------ F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 710 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 198049 5 71 3 2 G(n) = ------ F(n) F(n + 1) + -- F(n) F(n + 1) - 41/7 F(n) 84 84 110473 3 3 2 8 22945 2 4 + ------ F(n) F(n + 1) + 5475/4 F(n) F(n + 1) - ----- F(n) F(n + 1) 84 14 5153 11 3 27 2 2 + ---- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 84 42 28 9 3 7 28073 2 - 46700/3 F(n) F(n + 1) + 36325/3 F(n) F(n + 1) - ----- F(n + 1) - 1/7 84 4 74875 10 124013 6 + 1/7 F(n + 1) + ----- F(n + 1) - ------ F(n + 1) 12 21 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 711 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1714033 6 4160151 9 11459787 5 G(n) = -------- F(n + 1) + ------- F(n) F(n + 1) + -------- F(n) F(n + 1) 111650 22330 55825 73 4 983537 9 5517459 8 2 + -- F(n) F(n + 1) - ------ F(n) F(n + 1) - ------- F(n) F(n + 1) 11 4466 11165 207 3 2 2 3 11293071 10 - --- F(n) F(n + 1) + 7/2 F(n) F(n + 1) - -------- F(n + 1) 22 111650 6500052 2 269 2041 2967638 2 + ------- F(n + 1) - --- F(n) + ---- F(n + 1) + ------- F(n) 55825 638 638 55825 596191 10 3173684 2 4 2447273 - ------ F(n) + ------- F(n) F(n + 1) + ------- F(n) F(n + 1) 10150 11165 55825 69 5 - -- F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 712 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 162 8 42 4 981 357 3 G(n) = ----- F(n) + ---- F(n) + --- F(n) + --- F(n) F(n + 1) 1595 1595 319 638 66657 3 2 1077 3 178115 2 7 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) 638 3190 638 201 2 6 406355 5 4 39 5 3 + --- F(n) F(n + 1) + ------ F(n) F(n + 1) - -- F(n) F(n + 1) 638 638 29 17210 4 5 525 4 4 1944 3 5 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 638 1595 1180 4 86877 2 3 10145 4 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 319 14565 3 6 105 8 105 9 - ----- F(n) F(n + 1) - --- F(n + 1) + --- F(n + 1) 22 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 713 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2749119 10 264181 6 125 3 G(n) = -------- F(n + 1) - ------ F(n + 1) - --- F(n) F(n + 1) + 5/14 19600 9800 84 1683289 10 1558689 2 3277481 2 4 - ------- F(n) + ------- F(n) + ------- F(n + 1) - 5/14 F(n + 1) 19600 19600 19600 15 2 2 1531007 2 4 2920093 5 + -- F(n) F(n + 1) + ------- F(n) F(n + 1) + ------- F(n) F(n + 1) 28 3920 9800 1517671 9 25 3 2584227 8 2 + ------- F(n) F(n + 1) + -- F(n) F(n + 1) - ------- F(n) F(n + 1) 5880 42 3920 199651 9 288411 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 588 4900 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 714 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 54738 3 7635225 11 130770518 7 G(n) = ----- F(n) + ------- F(n + 1) - --------- F(n + 1) 6061 319 6061 84300479 6 19179400 10 189 + -------- F(n) F(n + 1) - -------- F(n) F(n + 1) - ---- 12122 319 6061 2118 3 14474300 3 8 3520322 2 + ---- F(n) F(n + 1) + -------- F(n) F(n + 1) + ------- F(n) F(n + 1) 6061 319 6061 2098075 2 9 4461 3 14298067 3 + ------- F(n) F(n + 1) + ----- F(n) F(n + 1) - -------- F(n + 1) 319 12122 6061 501 4 86047957 2 5 1467 2 2 - ---- F(n + 1) - -------- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 12122 1102 613442 2 47449316 3 4 - ------ F(n) F(n + 1) + -------- F(n) F(n + 1) 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 715 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 103494 2 3355557 6 661638 5 G(n) = ------- F(n + 1) - ------- F(n + 1) + ------ F(n) F(n + 1) 319 638 319 5 4023 2 61425 10 39451 + 3/2 F(n + 1) - ---- F(n) + ----- F(n + 1) + ----- F(n) F(n + 1) 638 11 638 195 531 28525 2 8 36 4 + --- F(n) - --- F(n + 1) + ----- F(n) F(n + 1) - -- F(n) F(n + 1) 638 319 22 11 3 2 479461 2 4 39 2 3 + 9/2 F(n) F(n + 1) - ------ F(n) F(n + 1) - -- F(n) F(n + 1) 319 22 306975 9 399398 3 3 237125 3 7 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 22 319 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 716 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 8 10 4 6 G(n) = 1/84 F(n) (719 F(n) - 745 F(n) + 782 F(n) + 77 F(n) F(n + 1) 5 3 6 4 7 + 1438 F(n) F(n + 1) + 42453 F(n) F(n + 1) - 1788 F(n) F(n + 1) 7 3 8 2 9 - 52496 F(n) F(n + 1) + 5042 F(n) F(n + 1) - 5625 F(n) F(n + 1) 8 10 3 5 - 26 F(n + 1) + 14 F(n + 1) + 104 F(n) F(n + 1) - 70 F(n) F(n + 1) 2 4 2 6 3 + 518 F(n) F(n + 1) - 19 F(n) F(n + 1) + 421 F(n) F(n + 1) 3 7 4 2 4 4 - 2590 F(n) F(n + 1) + 11921 F(n) F(n + 1) - 130 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 717 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(4 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5985 11 243 10 105 11 126135 6 5 G(n) = ---- F(n) - --- F(n) + --- F(n + 1) - ------ F(n) F(n + 1) 638 638 638 638 603 6 4 119545 9 2 873 9 + --- F(n) F(n + 1) + ------ F(n) F(n + 1) + --- F(n) F(n + 1) 319 638 638 8495 8 3 567 8 2 97609 7 4 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) + ----- F(n) F(n + 1) 29 319 319 1743 7 3 2903 4 3 36935 5 6 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 638 638 4803 5 5 41885 10 851 2 9 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - --- F(n) F(n + 1) 638 638 638 1821 4 6 283 6 1685 2 5 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 638 1041 2 4 267 2 8 525 9 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 638 638 638 105 10 - --- F(n + 1) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 718 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 3 2 G(n) = 85/2 F(n) F(n + 1) + 11/6 F(n) - 19/2 F(n) F(n + 1) 4 2 5 2 + 8 F(n) F(n + 1) - 13 F(n) F(n + 1) - 12 F(n) F(n + 1) 2 3 3 4 6 + 3/2 F(n) F(n + 1) + 481/2 F(n) F(n + 1) - 259 F(n) F(n + 1) 3 5 - 217/2 F(n + 1) - 5/6 F(n) + 7/2 F(n + 1) - 7/2 F(n + 1) 7 + 217/2 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 719 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(n + j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 874 7 223 2 2 27 2 6 G(n) = --- F(n) F(n + 1) + --- F(n) F(n + 1) + -- F(n) F(n + 1) 55 55 55 4 40 2 3 254 3 + 9/11 F(n) F(n + 1) - -- F(n) F(n + 1) - --- F(n) F(n + 1) 11 55 46 3 2 41 4 28 8 + -- F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) - 5/22 F(n + 1) 11 22 11 5 56 5 3 6 2 + 7/22 F(n) + 5/22 + -- F(n) F(n + 1) - 35/2 F(n) F(n + 1) 11 15 3 - -- F(n) F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 720 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(n + j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 3 5 2 G(n) = -1/6 F(n) (18 F(n) + 18 F(n) F(n + 1) + 486 F(n) F(n + 1) 3 4 3 2 2 5 + 555 F(n) F(n + 1) - 27 F(n) F(n + 1) - 270 F(n) F(n + 1) 6 7 3 5 - 660 F(n) F(n + 1) + 333 F(n + 1) - 337 F(n + 1) - 6 F(n) 4 2 + 21 F(n) F(n + 1) - 131 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 721 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6493 4 3260 2 7 47 5 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + -- F(n) F(n + 1) 44 11 44 61 5 481 3 3 115 - -- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 44 44 44 180 4 5 4 2 7045 4 - --- F(n) F(n + 1) - 27/2 F(n) F(n + 1) - ---- F(n) F(n + 1) 11 22 3 6 11313 3 2 14145 5 4 - 715 F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 44 22 17 6 6239 5 9 6405 2 - -- F(n) - ---- F(n) + 5/22 F(n + 1) + ---- F(n) - 5/22 F(n + 1) 22 44 44 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 722 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 2 4 G(n) = 1/90 F(n) (6397 F(n + 1) - 6861 F(n) F(n + 1) - 1350 F(n) F(n + 1) 8 2 2 6 - 3372 F(n + 1) + 660 F(n + 1) + 555 F(n) - 720 F(n) 5 3 7 6 2 - 25664 F(n) F(n + 1) + 9250 F(n) F(n + 1) + 12425 F(n) F(n + 1) 6 7 - 675 F(n + 1) - 525 F(n) F(n + 1) + 7555 F(n) F(n + 1) 5 8 + 1890 F(n) F(n + 1) + 3385 F(n) - 2950) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 723 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(n + 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 597 2 3 207303 2 68938 2 208183 6 G(n) = ---- F(n) F(n + 1) + ------ F(n + 1) + ----- F(n) - ------ F(n + 1) 638 14036 3509 14036 1715789 7 3 95 8 2037007 8 2 + ------- F(n) F(n + 1) + --- F(n) F(n + 1) - ------- F(n) F(n + 1) 7018 638 14036 146799 9 90267 10 128071 3 3 - ------ F(n) F(n + 1) - ----- F(n) - ------ F(n) F(n + 1) 3509 3509 7018 15 4 25 9 38 30 3 6 - -- F(n) F(n + 1) + --- F(n) - --- F(n + 1) + --- F(n) F(n + 1) 29 319 319 319 205587 9 25 8 9439 2 4 + ------ F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 7018 319 3509 829589 2 8 125 2 7 177 3 2 - ------ F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 14036 638 319 5 + 2/11 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 724 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(n + 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 13834 5 9 2 7 G(n) = ------ F(n + 1) + 2605/2 F(n + 1) + 280 F(n) F(n + 1) 11 4589 3 2 6016 4 3 3 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + 56 F(n) F(n + 1) 22 11 3 6 159 8 + 5065/2 F(n) F(n + 1) + --- F(n) F(n + 1) - 6495/2 F(n) F(n + 1) 22 491 58 7177 2 3 2 4 - --- F(n + 1) + -- F(n) - ---- F(n) F(n + 1) - 11 F(n) F(n + 1) 11 11 22 14 2 489 2 6 5 - -- F(n) - --- F(n + 1) + 22 F(n + 1) - 101/2 F(n) F(n + 1) 11 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 725 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 60479 5 95 6 510 5 2 G(n) = ----- F(n) F(n + 1) + -- F(n) F(n + 1) + --- F(n) F(n + 1) 22 22 11 30255 9 134 2 75210 2 8 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) + ----- F(n) F(n + 1) 11 11 11 22801 3 3 185 3 4 71 2 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) - -- F(n) F(n + 1) 11 11 22 118795 4 6 29281 2 35 3 14557 6 - ------ F(n) F(n + 1) - ----- F(n) + -- F(n) + ----- F(n) 22 22 22 11 5770 3 7 34149 4 2 4 3 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - 20 F(n) F(n + 1) 11 22 59863 5 7 10 + ----- F(n) F(n + 1) - 5/22 F(n + 1) + 5/22 F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 726 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 643931 6 4 16799 2 G(n) = ------ F(n) F(n + 1) + 23/6 F(n) F(n + 1) + ----- F(n) F(n + 1) 84 28 2 3 2913 2 10 - 7/12 F(n) F(n + 1) - ---- F(n) F(n + 1) - 389075/6 F(n) F(n + 1) 28 588925 3 8 115043 3 4 3 2 + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - 13/4 F(n) F(n + 1) 12 14 1963289 7 199 3 10 17 17 5 - ------- F(n + 1) + --- F(n) - -- F(n) + -- F(n + 1) - -- F(n + 1) 84 21 21 12 12 51634 3 11 83575 2 9 - ----- F(n + 1) + 103325/4 F(n + 1) + ----- F(n) F(n + 1) 21 12 212351 2 5 - ------ F(n) F(n + 1) 28 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 727 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(n + 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3181 7 310 6 346311 4 7 471 4 6 G(n) = ---- F(n) - --- F(n) + ------ F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 319 697553 4 3 4 2 793632 5 6 - ------ F(n) F(n + 1) - 7/22 F(n) F(n + 1) - ------ F(n) F(n + 1) 638 319 471 5 5 1162169 5 2 853082 7 4 + --- F(n) F(n + 1) + ------- F(n) F(n + 1) + ------ F(n) F(n + 1) 638 3190 319 471 7 3 27177 6 5 471 6 4 - --- F(n) F(n + 1) - ----- F(n) F(n + 1) - --- F(n) F(n + 1) 638 58 319 61 2 257 239 3 8 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 110 3190 638 471 3 7 2689 5 2 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) + 3/145 F(n + 1) 638 1595 937 3 67 7 26768 6 67 6 - ---- F(n + 1) + --- F(n + 1) - ----- F(n) F(n + 1) + ---- F(n + 1) 3190 290 319 1595 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 728 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 2 2 G(n) = -1/6 F(n) (166 F(n + 1) + 5 F(n) - 29 - 952 F(n) F(n + 1) 4 8 3 - 6111 F(n + 1) + 6135 F(n + 1) - 46 F(n) F(n + 1) + 2547 F(n) F(n + 1) 7 2 4 3 - 14865 F(n) F(n + 1) + 105 F(n) F(n + 1) + 295 F(n) F(n + 1) 3 5 6 5 + 13080 F(n) F(n + 1) - 165 F(n + 1) + 375 F(n) F(n + 1) 2 6 3 3 - 105 F(n) F(n + 1) - 435 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 729 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(2 n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1196 7 3 6 25 7 57 7 G(n) = ---- F(n) F(n + 1) - 1/2 F(n + 1) + -- F(n + 1) + -- F(n) 11 44 22 111 5 813 4 6 189 10 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) 22 22 22 91 2 8 669 3 7 689 4 3 + -- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 11 22 44 309 2 5 2 823 9 - --- F(n) F(n + 1) + 8/11 F(n + 1) + --- F(n) F(n + 1) 44 22 602 6 4 6 148 3 4 - --- F(n) F(n + 1) - 9 F(n) F(n + 1) + --- F(n) F(n + 1) 11 11 167 5 2 35 3 2249 8 2 + --- F(n) F(n + 1) - -- F(n + 1) - ---- F(n) F(n + 1) 11 44 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 730 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(2 n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 970939 6 160219 10 G(n) = -335 F(n + 1) + ------ F(n + 1) + 11/6 - ------ F(n + 1) 2040 680 258533 2 80851 10 3 - ------ F(n) + ----- F(n) - 425/3 F(n) F(n + 1) 2040 680 2 2 28735 9 7 + 329/6 F(n) F(n + 1) + ----- F(n) F(n + 1) + 815 F(n) F(n + 1) 34 42673 2 4 32103 9 3 5 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - 705 F(n) F(n + 1) 136 68 104417 2 8 79963 5 2 6 - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) - 15/2 F(n) F(n + 1) 136 85 3 88901 4 - 35/2 F(n) F(n + 1) + ----- F(n) F(n + 1) + 1999/6 F(n + 1) 1020 245141 2 - ------ F(n + 1) 1020 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 731 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 n - j) F(2 n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 1259 3 72 2 G(n) = -4637/4 F(n) F(n + 1) + ---- F(n) F(n + 1) - -- F(n) F(n + 1) 110 11 2247 6 371 8 136 7 137 7 - ---- F(n) F(n + 1) - --- F(n) + --- F(n) + --- F(n + 1) 22 44 11 22 284 4 189 3 75 11 29 8 - --- F(n + 1) - --- F(n + 1) - -- F(n + 1) - --- F(n + 1) 55 44 44 220 62775 5 6 5 3 9323 5 2 - ----- F(n) F(n + 1) + 358/5 F(n) F(n + 1) + ---- F(n) F(n + 1) 22 22 27425 4 7 1135 4 4 30200 7 4 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 22 44 11 35 7 13425 6 5 521 6 2 223 - -- F(n) F(n + 1) - ----- F(n) F(n + 1) - --- F(n) F(n + 1) + --- 11 44 11 44 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 732 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 n - j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 5 G(n) = 1/4200 F(n) (-2874200 F(n + 1) + 6195000 F(n) F(n + 1) 7 10 - 6982500 F(n) F(n + 1) + 2839639 F(n + 1) - 15400 2 6 3 - 105000 F(n) F(n + 1) - 1037664 F(n) F(n + 1) + 1180900 F(n) F(n + 1) 8 2 10 + 2887500 F(n + 1) - 1804209 F(n) + 1857409 F(n) 9 6 3 + 7870200 F(n) F(n + 1) + 796122 F(n + 1) + 139300 F(n) F(n + 1) 2 8 2 9 - 3630861 F(n + 1) + 12604435 F(n) F(n + 1) - 5211170 F(n) F(n + 1) 2 2 2 4 - 441000 F(n) F(n + 1) - 7561635 F(n) F(n + 1) 5 - 6706866 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 733 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 185355 356481 356481 59251420 8 G(n) = ------- F(n + 1) + ------- F(n) - ------- + -------- F(n + 1) 2280938 2280938 2280938 103679 105291818 4 68815625 12 1105353 2 2 + --------- F(n + 1) - -------- F(n + 1) + ------- F(n) F(n + 1) 1140469 103679 207358 173340000 11 13838480 7 + --------- F(n) F(n + 1) - -------- F(n) F(n + 1) 103679 103679 30393767 3 28940240 3 5 - -------- F(n) F(n + 1) - -------- F(n) F(n + 1) 1140469 103679 182385 3 20456250 2 10 - ------- F(n) F(n + 1) - -------- F(n) F(n + 1) 1140469 103679 129197500 3 9 21303240 2 6 - --------- F(n) F(n + 1) + -------- F(n) F(n + 1) 103679 103679 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 734 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 2 9 G(n) = 1/792 F(n) (13680 F(n) F(n + 1) - 95 F(n) F(n + 1) 10 8 3 7 2 - 12456 F(n) F(n + 1) + 14201 F(n) F(n + 1) + 810 F(n) F(n + 1) 9 2 2 3 2 5 - 32112 F(n) F(n + 1) - 405 F(n) F(n + 1) - 1089 F(n) F(n + 1) 2 7 6 3 5 6 + 405 F(n) F(n + 1) + 2025 F(n) F(n + 1) - 2940 F(n) F(n + 1) 6 5 7 4 + 1260 F(n) F(n + 1) + 324 F(n + 1) - 360 F(n + 1) - 810 F(n) F(n + 1) 4 5 4 3 7 4 - 2025 F(n) F(n + 1) + 3615 F(n) F(n + 1) + 15972 F(n) F(n + 1) 11 5 + 162 F(n) - 162 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 735 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 484713 3 13075625 12 11258335 8 G(n) = ------- F(n) - -------- F(n + 1) + -------- F(n + 1) 2412278 7562 7562 503145 3 289848539 4 484713 + ------- F(n + 1) + --------- F(n + 1) - ------- 2412278 1206139 2412278 83563167 3 1842461 2 2 - -------- F(n) F(n + 1) + ------- F(n) F(n + 1) 1206139 126962 246897 2 16468125 11 + ------- F(n) F(n + 1) + -------- F(n) F(n + 1) 1206139 3781 1314720 7 450315 2 - ------- F(n) F(n + 1) + ------- F(n) F(n + 1) 3781 2412278 12274375 3 9 2749460 3 5 - -------- F(n) F(n + 1) - ------- F(n) F(n + 1) 3781 3781 1658181 3 3886875 2 10 - ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 1206139 7562 2023910 2 6 + ------- F(n) F(n + 1) 3781 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 736 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 2 G(n) = ) F(j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 5 G(n) = -3/308 F(n) (42 F(n + 1) - 21 F(n) + 21 F(n) - 350 F(n) F(n + 1) 7 2 6 + 1450 F(n + 1) + 537 F(n) F(n + 1) - 3425 F(n) F(n + 1) 2 3 3 4 - 146 F(n) F(n + 1) - 1458 F(n + 1) + 3350 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 737 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5275139 6 2 55131 7 404905 7 G(n) = ------- F(n) F(n + 1) - ----- F(n) F(n + 1) + ------- F(n) F(n + 1) 2412278 41591 2412278 1537857 2 393061 3 468783 7 + -------- F(n) F(n + 1) - ------- F(n) F(n + 1) - ------- F(n) 12061390 2412278 1206139 3479661 2 503145 3 503145 434421 4 + -------- F(n) F(n + 1) - ------- F(n + 1) + ------- + ------- F(n) 12061390 2412278 2412278 2412278 468783 6 2709973 3 5 - ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 6030695 2412278 937566 4 3 2606573 4 4 + ------- F(n) F(n + 1) + ------- F(n) F(n + 1) 1206139 1206139 4530377 5 3 5625396 6 - ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 2412278 6030695 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 738 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 2 5 3 G(n) = 3/40 F(n + 1) (938 F(n + 1) + 21 F(n) F(n + 1) + 26 F(n) 3 3 4 2 - 938 F(n + 1) + 1974 F(n) F(n + 1) - 141 F(n + 1) F(n) 2 6 + 402 F(n + 1) F(n) - 2282 F(n + 1) F(n)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 739 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 300279 7 3662520 3 185355 5 G(n) = ------- F(n) F(n + 1) - ------- F(n) F(n + 1) - ------- F(n + 1) 1140469 1140469 2280938 11464545 5 3 13967208 6 2 223956 - -------- F(n) F(n + 1) + -------- F(n) F(n + 1) + ------- F(n) 1140469 1140469 1140469 893394 5 688527 3 2 915720 7 - ------- F(n) - ------- F(n) F(n + 1) + ------- F(n) F(n + 1) 1140469 1140469 1140469 1519371 4 1971 4 629535 3 + ------- F(n) F(n + 1) - ----- F(n) F(n + 1) - ------- F(n) F(n + 1) 2280938 11462 1140469 698349 3 5 60858 8 185355 8 + ------- F(n) F(n + 1) + ------ F(n) + ------- F(n + 1) 1140469 103679 2280938 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 740 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 13703 6 3 59743 7 2 G(n) = -9/44 F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 3850 7700 9221 8 4 3 45 5 2 - ---- F(n) F(n + 1) + 9/11 F(n) F(n + 1) + -- F(n) F(n + 1) 700 44 189 6 7141 3 2 231303 5 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) + ------ F(n) 44 3850 38500 193 5 1737 7787 81 3 + ---- F(n + 1) - ----- F(n + 1) + ----- F(n) - -- F(n + 1) 1375 19250 38500 44 11167 9 193 9 81 7 117 3 4 - ----- F(n) - ---- F(n + 1) + -- F(n + 1) + --- F(n) F(n + 1) 1925 3850 44 44 7 - 9/22 F(n) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 741 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 186113424 5 952092 4 4129119 3 G(n) = --------- F(n + 1) - ------- F(n + 1) - ------- F(n) F(n + 1) 1206139 1206139 2412278 42717400 8 3801753 3 + -------- F(n) F(n + 1) + ------- F(n) F(n + 1) 109649 2412278 765911125 3 6 24732975 2 7 - --------- F(n) F(n + 1) - -------- F(n) F(n + 1) 2412278 1206139 41111724 2 3 191745 2 2 1401039 + -------- F(n) F(n + 1) - ------- F(n) F(n + 1) + ------- 1206139 2412278 2412278 1401039 4578861 159789169 4 - ------- F(n) + ------- F(n + 1) - --------- F(n) F(n + 1) 2412278 1206139 2412278 22459927 3 2 380881425 9 - -------- F(n) F(n + 1) - --------- F(n + 1) 1206139 2412278 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 742 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 3 G(n) = 1/40 F(n + 1) (-2375 F(n) F(n + 1) + 925 F(n) F(n + 1) 3 8 2 6 + 5235 F(n) F(n + 1) + 11760 F(n + 1) - 114 + 1475 F(n) F(n + 1) 7 3 5 4 - 28995 F(n) F(n + 1) + 23735 F(n) F(n + 1) - 11646 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 743 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 3 3 G(n) = -1/1540 F(n) (1260 F(n + 1) - 1146 F(n) F(n + 1) + 945 F(n) 7 2 8 - 1890 F(n) + 2520 F(n) F(n + 1) + 7093 F(n) F(n + 1) 6 8 - 1890 F(n) F(n + 1) - 750 F(n + 1) - 2430 F(n) F(n + 1) 7 2 6 3 5 4 + 11455 F(n) F(n + 1) - 17954 F(n) F(n + 1) + 24710 F(n) F(n + 1) 6 2 5 9 - 3780 F(n) F(n + 1) + 3780 F(n) F(n + 1) + 945 F(n) 2 7 4 - 16150 F(n) F(n + 1) - 6718 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 744 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1528392067 6 8304453 5 42475 2 8 G(n) = ---------- F(n + 1) - ------- F(n + 1) - ----- F(n) F(n + 1) 2412278 2412278 319 7544337 2338605 2 528700 9 - ------- F(n) F(n + 1) + ------- F(n) + ------ F(n) F(n + 1) 1206139 2412278 319 307632064 5 9716076 4 - --------- F(n) F(n + 1) + ------- F(n) F(n + 1) 1206139 1206139 415400 3 7 156366682 3 3 - ------ F(n) F(n + 1) - --------- F(n) F(n + 1) 319 1206139 17623989 3 2 406687769 2 4 - -------- F(n) F(n + 1) + --------- F(n) F(n + 1) 2412278 2412278 44928 2 3 7325 10 2338605 - ------ F(n) F(n + 1) - ---- F(n + 1) - ------- F(n) 109649 11 2412278 3900654 39234464 2 + ------- F(n + 1) + -------- F(n + 1) 1206139 1206139 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 745 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 3 7 G(n) = 1/308 F(n) (10753 F(n) F(n + 1) + 147125 F(n) F(n + 1) 3 5 3 3 3 - 630 F(n) F(n + 1) - 25829 F(n) F(n + 1) + 945 F(n) F(n + 1) 2 8 2 6 2 4 - 412500 F(n) F(n + 1) + 315 F(n) F(n + 1) + 59502 F(n) F(n + 1) 6 4 6 4 - 69912 F(n + 1) + 361625 F(n) F(n + 1) - 315 F(n) 4 4 8 10 - 315 F(n) F(n + 1) - 315 F(n + 1) + 70125 F(n + 1) 3 5 - 1653 F(n) F(n + 1) + 630 F(n) F(n + 1) - 139236 F(n) F(n + 1) 7 2 2 6 + 630 F(n) F(n + 1) - 1260 F(n) F(n + 1) + 315 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 746 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 46101261 5 7 22489689 7 G(n) = --------- F(n) F(n + 1) + -------- F(n) F(n + 1) 2828188 2828188 31840773 6 6 22888361 11 + -------- F(n) F(n + 1) - -------- F(n) F(n + 1) 707047 2828188 68319 2 6945 6 5 27927 5 2 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 166364 83182 166364 166048 3 21005782 7 5 3627 6 - ------ F(n) F(n + 1) - -------- F(n) F(n + 1) - ---- F(n) F(n + 1) 321385 707047 7562 11418719 6 2 6945 10 - -------- F(n) F(n + 1) + ------ F(n) F(n + 1) 1414094 166364 129141977 5 3 34725 5 6 190681 + --------- F(n) F(n + 1) - ----- F(n) F(n + 1) - ------ 14140940 41591 707047 22608 6 173625 7 4 34725 8 3 - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 41591 166364 83182 1305256 4 12003 3 15663 7 20835 7 + ------- F(n + 1) - ----- F(n + 1) + ----- F(n + 1) - ----- F(n) 3535235 83182 83182 83182 735557 12 41058 12 41923 8 + ------- F(n) + ------ F(n + 1) - ------ F(n + 1) 1414094 707047 207955 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 747 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 n - 3 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 7526179 4 2463125 12 3258 20697 G(n) = ------- F(n + 1) - ------- F(n + 1) - ---- F(n) + ----- F(n + 1) 12122 551 6061 12122 3258 17080 7 2031 4 + ---- - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 6061 19 551 1035860 3 5 5475 3 2 46162 3 - ------- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 551 1102 6061 1464375 2 10 762510 2 6 2 3 - ------- F(n) F(n + 1) + ------ F(n) F(n + 1) + 9/19 F(n) F(n + 1) 1102 551 1070776 3 23139 2 2 6204375 11 - ------- F(n) F(n + 1) + ----- F(n) F(n + 1) + ------- F(n) F(n + 1) 6061 551 551 4624375 3 9 2007 5 4241585 8 - ------- F(n) F(n + 1) - ---- F(n + 1) + ------- F(n + 1) 551 1102 1102 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 748 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 690 3 5019 3 5000 2 6 301918 4 G(n) = ----- F(n + 1) - ----- F(n) - ---- F(n) F(n + 1) + ------ F(n + 1) 6061 12122 6061 6061 127960 3 333 2 50624 2 2 - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 12122 6061 720 2 17008 3 55225 8 5019 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n + 1) + ----- 6061 6061 1102 12122 38900 7 637800 3 5 + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 319 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 749 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 n - 3 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 382075 3 5 45849 4 1830 4695 4 G(n) = ------- F(n) F(n + 1) + ----- F(n) - ----- + ---- F(n) F(n + 1) 41591 83182 41591 2189 169621 3 318 3 2 7819125 2 6 - ------ F(n) F(n + 1) - --- F(n) F(n + 1) - ------- F(n) F(n + 1) 41591 199 83182 1237285 2 2 4881 2 3 3849025 4 4 + ------- F(n) F(n + 1) - ---- F(n) F(n + 1) + ------- F(n) F(n + 1) 83182 4378 41591 1461 4 1666717 3 1668225 7 + ---- F(n) F(n + 1) - ------- F(n) F(n + 1) + ------- F(n) F(n + 1) 4378 41591 41591 42189 1830 - ----- F(n) + ----- F(n + 1) 83182 41591 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 750 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 n - 3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 427 5 1129 3 3 19 91 6 G(n) = ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) 99 198 33 66 169 4 26 4 988 3 5 2873 6 2 - --- F(n) + -- F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 198 33 99 198 1235 5 3 1079 4 4 4 2 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + 47/6 F(n) F(n + 1) 99 66 19 2 117 7 26 8 19 6 52 - --- F(n + 1) + --- F(n) F(n + 1) - -- F(n + 1) + --- F(n + 1) - -- 396 11 99 396 99 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 751 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 n - 3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5235 3 5033 8 1233 G(n) = ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- 12122 836 6061 377945 4 2055 2 2 160449 2 3 + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) 24244 6061 30305 1442679 2 7 14613 3 497733 3 2 + ------- F(n) F(n + 1) - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 121220 12122 121220 2362067 7 2 61455 8 117565 - ------- F(n) F(n + 1) - ----- F(n) F(n + 1) + ------ F(n + 1) 121220 24244 24244 1233 9 114805 5 1923 4 - ---- F(n) - ------ F(n + 1) - ---- F(n + 1) 6061 24244 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 752 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 n - 3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 39438 2 2234773 1456164 5945371 9 G(n) = ------- F(n + 1) - ------- F(n + 1) + ------- F(n) + ------- F(n) 207955 1039775 1039775 415910 433871 9 753264 5 17233279 5 + ------- F(n + 1) - ------ F(n) F(n + 1) - -------- F(n) 1039775 207955 1039775 1755152 5 25164 2901483 5 4 + ------- F(n + 1) + ----- F(n) F(n + 1) - ------- F(n) F(n + 1) 1039775 18905 1039775 9777 2 4 48588 6 6645 6 + ---- F(n) F(n + 1) + ------ F(n + 1) + ---- F(n) 4378 207955 7562 2860467 8 4815913 7 2 80847 5 + ------- F(n) F(n + 1) - ------- F(n) F(n + 1) - ----- F(n) F(n + 1) 83182 207955 41591 6693352 4 - ------- F(n) F(n + 1) 1039775 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 753 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 n - 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 26775 2309677 9 624301 8 2 G(n) = ------ F(n) - ------- F(n) F(n + 1) + ------ F(n) F(n + 1) 48488 48488 24244 26865 8 596949 7 3 97335 7 2 - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 96976 48488 48488 251569 4 6 330975 4 5 + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 193952 96976 1705239 4 2 17763 4 53033 9 + ------- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 193952 8816 96976 6219 8 31267 5 1071 4 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 48488 48488 1276 4199383 10 3165 9 4319143 6 398391 6 3 - ------- F(n) - ----- F(n) + ------- F(n) - ------ F(n) F(n + 1) 193952 48488 193952 96976 35053 690 9 690 10 - ----- F(n) F(n + 1) - ---- F(n + 1) + ---- F(n + 1) 96976 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 754 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 n - 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 690 11 1380 9 2047985 3 4 G(n) = ---- F(n + 1) + ---- F(n) F(n + 1) - ------- F(n) F(n + 1) 6061 6061 6061 8501120 10 9951 3 9951 2 + ------- F(n) F(n + 1) - ----- F(n) + ----- F(n) 6061 12122 12122 16299715 6 690 10 26478 3 3 - -------- F(n) F(n + 1) - ---- F(n + 1) - ----- F(n) F(n + 1) 12122 6061 6061 21095440 2 9 4899 2 4 114645 2 5 - -------- F(n) F(n + 1) - ---- F(n) F(n + 1) + ------ F(n) F(n + 1) 6061 551 209 690 2 8 49377 5 122271 2 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 6061 12122 12122 2334 352291 2 16808565 4 7 - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) + -------- F(n) F(n + 1) 551 6061 6061 690 4 6 2946245 4 3 123 4 2 - ---- F(n) F(n + 1) + ------- F(n) F(n + 1) + --- F(n) F(n + 1) 6061 12122 11 1380 3 7 1490505 3 8 - ---- F(n) F(n + 1) + ------- F(n) F(n + 1) 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 755 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 32515 4 8 6480 3 1977 12 1977 9 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) + ----- F(n) 6061 6061 12122 12122 615 9 1245 8 936563 8 4 + ----- F(n + 1) - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 12122 418 12122 4750 5 3 407388 7 373782 11 - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 319 6061 6061 78690 6 6 103929 7 2 17601 3 6 + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 12122 6061 50403 4 25588 9 3 934974 10 2 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 12122 319 6061 36626 3 5 615 12 123 2 10 + ----- F(n) F(n + 1) - ----- F(n + 1) - --- F(n) F(n + 1) 6061 12122 209 3609 2 7 879 6 3 11241 4 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 12122 12122 12122 118527 5 4 + ------ F(n) F(n + 1) 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 756 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 n - 2 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 633 6 33 633 2 G(n) = -111/2 F(n) F(n + 1) - --- F(n + 1) - -- + --- F(n + 1) 28 28 28 14881 4 3 5 117 2 4 - ----- F(n + 1) + 3220 F(n) F(n + 1) + --- F(n) F(n + 1) 14 14 2 6 31875 2 10 2025 2 2 - 2370 F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 14 28 4241 3 99 11 + ---- F(n) F(n + 1) - -- F(n) F(n + 1) - 135000/7 F(n) F(n + 1) 14 14 7 5 3 + 1540 F(n) F(n + 1) + 363/7 F(n) F(n + 1) + 29/2 F(n) F(n + 1) 3 9 8 33 2 + 14375 F(n) F(n + 1) - 46145/7 F(n + 1) + -- F(n) 28 12 + 30625/4 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 757 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 n - 2 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5265 4 2940 615 615 4 G(n) = ----- F(n) - ---- F(n) - ----- F(n + 1) + ----- + 9/22 F(n) F(n + 1) 12122 6061 12122 12122 142353 2 2 15 2 3 19521 3 + ------ F(n) F(n + 1) - -- F(n) F(n + 1) - ----- F(n) F(n + 1) 12122 11 6061 3 2 51 4 191709 3 - 3/2 F(n) F(n + 1) + -- F(n) F(n + 1) - ------ F(n) F(n + 1) 22 6061 191925 7 899325 2 6 442800 4 4 + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 6061 12122 6061 43875 3 5 - ----- F(n) F(n + 1) 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 758 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 n - 2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 416 8 32 640 4 4096 7 G(n) = --- F(n + 1) + -- - --- F(n + 1) + ---- F(n) F(n + 1) 693 99 693 693 512 4 4 1163 3 3 2432 2 2 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 33 66 693 2 4 443 5 8 127 2 + 8 F(n) F(n + 1) + --- F(n) F(n + 1) - 16/9 F(n) - --- F(n) 66 22 49 2 159 6 49 6 20096 5 3 + --- F(n + 1) + --- F(n) - --- F(n + 1) + ----- F(n) F(n + 1) 132 22 132 693 8768 6 2 - ---- F(n) F(n + 1) 693 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 759 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 n - 2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 73185 6 64455 3 4 18225 2 6 G(n) = ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 638 7562 1519371 4 24467859 2 42714 3 + ------- F(n + 1) - -------- F(n) F(n + 1) - ----- F(n) F(n + 1) 7562 1206139 3781 1659798 3 113607435 3 321084 3 - ------- F(n) + --------- F(n + 1) - ------ F(n) F(n + 1) 1206139 2412278 3781 97875 7 8093532 2 127413 2 2 + ----- F(n) F(n + 1) + ------- F(n) F(n + 1) + ------ F(n) F(n + 1) 199 1206139 3781 1275 2 5 1601775 3 5 764100 8 - ---- F(n) F(n + 1) - ------- F(n) F(n + 1) - ------ F(n + 1) 638 3781 3781 1659798 15090 7 + ------- - ----- F(n + 1) 1206139 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 760 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 n - 2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 93 5 111 2 4 4 2 G(n) = -- F(n) F(n + 1) - --- F(n) F(n + 1) + 12 F(n) F(n + 1) 22 11 247035 4 5 53706 4 47845 3 6 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 6061 12122 42 3 3 169657 3 2 2651645 2 7 - -- F(n) F(n + 1) - ------ F(n) F(n + 1) - ------- F(n) F(n + 1) 11 6061 6061 453161 2 3 99660 8 2183289 4 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ------- F(n) F(n + 1) 6061 551 12122 831 2 27141 831 615 2 + ---- F(n) - ----- F(n) F(n + 1) - ---- F(n) - ----- F(n + 1) 1102 6061 1102 12122 615 9 + ----- F(n + 1) 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 761 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 n - 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 689 7 2 7179 8 1829 3 6 G(n) = --- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 21 616 132 485 2 7 3547 4 5 681 5 2 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 264 132 44 97 853 2 5 2171 9 2171 7 - --- F(n + 1) + --- F(n) F(n + 1) + ---- F(n) - ---- F(n) 264 264 924 924 1709 5 4 8091 4 3 5959 6 3 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 44 616 132 4223 6 3089 3 4 853 3 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n + 1) 462 308 1848 97 5 853 7 + --- F(n + 1) - ---- F(n + 1) 264 1848 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 762 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 n - 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 84693 525 6 4747998 5 G(n) = ------ F(n) F(n + 1) + --- F(n) F(n + 1) - ------- F(n) F(n + 1) 6061 22 6061 47085 2 8775 9 4048289 2 4 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ------- F(n) F(n + 1) 6061 11 12122 965311 3 3 43075 2 8 1245 2 5 - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 22 22 1245 4 3 959657 4 2 800 3 7 + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) + --- F(n) F(n + 1) 22 12122 11 17950 4 6 615 3 90 3 4 + ----- F(n) F(n + 1) - ----- F(n + 1) - -- F(n) F(n + 1) 11 12122 11 15021 3 15021 2 147216 2 615 2 - ----- F(n) + ----- F(n) - ------ F(n) F(n + 1) + ----- F(n + 1) 12122 12122 6061 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 763 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 41565 7 20730 3 502542 4 G(n) = ------ F(n + 1) + ----- F(n + 1) + ------ F(n + 1) 638 319 319 3345 2 5 1120840 2 6 - ---- F(n) F(n + 1) + ------- F(n) F(n + 1) 638 319 1076250 2 10 7529 3 86235 3 4 - ------- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 638 1522640 3 5 6797500 3 9 18255 2 - ------- F(n) F(n + 1) - ------- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 638 143007 3 9120000 11 219 2 - ------ F(n) F(n + 1) + ------- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 22 69547 2 2 50850 6 728080 7 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 638 319 319 1431 3 3117420 8 1431 3620625 12 - ---- F(n) + ------- F(n + 1) + ---- - ------- F(n + 1) 638 319 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 764 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 294 5 650 2 6 80875 8 G(n) = --- - 3 F(n + 1) - --- F(n) F(n + 1) - ----- F(n + 1) 319 319 638 1809 294 84900 3 5 17032 3 + ---- F(n + 1) - --- F(n) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 319 319 319 98400 7 2 3 81 3 2 + ----- F(n) F(n + 1) + 9/22 F(n) F(n + 1) - -- F(n) F(n + 1) 319 11 153 4 6740 2 2 2264 3 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 22 319 319 40196 4 + ----- F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 765 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 3 2 G(n) = 1/84 F(n) (14 F(n) F(n + 1) + 40 F(n + 1) + 658 F(n) F(n + 1) 4 5 2 2 - 378 F(n) F(n + 1) - 2250 F(n) F(n + 1) + 180 F(n) F(n + 1) 2 3 3 6 - 476 F(n) F(n + 1) - 28 F(n + 1) - 770 F(n) - 60 F(n) F(n + 1) 2 5 6 5 7 + 420 F(n) F(n + 1) + 1880 F(n) F(n + 1) + 105 F(n) + 665 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 766 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 103105 1161 3 147 3 18700 4 837 7 G(n) = ------ + ---- F(n) + ---- F(n + 1) - ----- F(n + 1) - ---- F(n + 1) 18183 209 6061 1653 6061 84327 7 3652 7 45100 3 - ----- F(n) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 12122 1653 1653 96250 6 2 2332 3 5 1083 2 5 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 1653 1653 638 43653 6 81153 3 4 32516 2 6 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 12122 1653 82503 4 3 14113 8 9515 8 + ----- F(n) F(n + 1) - ----- F(n) + ---- F(n + 1) 6061 3306 1653 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 767 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 819 2 7177 5 5827 9 105 9 6 G(n) = ---- F(n) - ---- F(n) + ---- F(n) + --- F(n + 1) + 9/44 F(n) 1276 1595 1595 638 147 4 2 63 2 4 22 8 + --- F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 44 44 29 764 2 3 6704 8 21985 5 4 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 638 42 5 12706 4 4 5 - -- F(n) F(n + 1) - ----- F(n) F(n + 1) + 104/5 F(n) F(n + 1) 11 1595 229 3 6 105 105 2 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n + 1) 319 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 768 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 n - j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 7 4 G(n) = -1/84 F(n) (38875 F(n + 1) - 94075 F(n) F(n + 1) - 38723 F(n + 1) 2 6 2 2 5 - 1175 F(n) F(n + 1) - 6000 F(n) F(n + 1) + 8575 F(n) F(n + 1) 3 5 3 3 - 1129 F(n) F(n + 1) + 83200 F(n) F(n + 1) - 9555 F(n) F(n + 1) 6 3 2 4 - 3745 F(n + 1) + 16021 F(n) F(n + 1) + 2065 F(n) F(n + 1) 3 2 2 + 1877 F(n) F(n + 1) + 3789 F(n + 1) + 208 F(n) - 208) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 769 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3587 5 11208 5 2841 675 4 4221 4 G(n) = ----- F(n + 1) - ----- F(n) + ----- + --- F(n) - ----- F(n + 1) 12122 6061 12122 418 12122 152580 4 4 687668 4 5 2207 - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) - ----- F(n + 1) 6061 6061 12122 14145 2 6 2319 2 7 98062 4 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 638 6061 33998 3 2 126405 3 5 328839 3 6 - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 551 12122 6061 103815 5 3 414914 5 4 486135 6 2 - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 6061 6061 12122 1072393 6 3 79074 3 + ------- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 770 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 n - j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 26049 4 2 1245 4 3 94655 4 6 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 22 58 90 3 4 254370 9 4893 2 - -- F(n) F(n + 1) + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 11 319 638 212161 2 4 1245 2 5 624640 2 8 + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 638 22 319 51044 3 3 564 2 23305 3 7 - ----- F(n) F(n + 1) + --- F(n) + ----- F(n) F(n + 1) 319 319 319 4882 249772 5 525 6 - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 22 105 3 7518 2 564 3 105 10 - --- F(n + 1) - ---- F(n) F(n + 1) - --- F(n) + --- F(n + 1) 638 319 319 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 771 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 5 6 G(n) = 1/420 F(n) (-5030 F(n) + 1565 F(n) + 3465 F(n) + 1088 F(n) F(n + 1) 2 8 7 2 - 38 F(n) F(n + 1) + 22570 F(n) F(n + 1) - 12710 F(n) F(n + 1) 6 3 6 5 4 + 11150 F(n) F(n + 1) - 1444 F(n) F(n + 1) - 82920 F(n) F(n + 1) 5 2 2 8 + 25650 F(n) F(n + 1) - 5706 F(n) F(n + 1) - 3190 F(n) F(n + 1) 4 3 7 4 - 22380 F(n) F(n + 1) - 300 F(n + 1) + 6600 F(n) F(n + 1) 9 4 5 + 360 F(n + 1) + 61270 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 772 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 n - j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 105 4 105 11 834 8 834 7 G(n) = ---- F(n + 1) + --- F(n + 1) + --- F(n) - --- F(n) 638 638 319 319 301328 7 4 79526 8 3 58572 9 2 - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 1595 1595 1595 199297 6 5 9861 3 5 12759 7 + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 1595 638 638 84873 10 309 3 1361 6 + ----- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 3190 29 110 4467 10 4617 2 2 59517 2 5 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 3190 638 1595 8229 2 6 20338 2 3447 3 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 1595 319 20433 2 225 7 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) 3190 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 773 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 105 105 3 63 2 53 3 G(n) = --- - --- F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 638 638 638 319 117 2 81 2 2 144 3 419 3 - --- F(n) F(n + 1) + -- F(n) F(n + 1) - --- F(n) - --- F(n) F(n + 1) 638 58 319 319 183 4 + --- F(n) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 774 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 690 690 6708 6074 3 G(n) = ---- - ---- F(n + 1) + ---- F(n) - ---- F(n) F(n + 1) 6061 6061 6061 6061 16011 2 2 393 2 3 19048 3 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 1102 6061 2529 3 2 12561 4 30 4 2487 5 + ---- F(n) F(n + 1) + ----- F(n) - -- F(n) F(n + 1) - ---- F(n) 1102 12122 29 1102 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 775 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 477 6 2 477 2 G(n) = -9/308 F(n + 1) + --- F(n + 1) - 9/308 F(n) - --- F(n + 1) + 9/308 77 77 195 3 27 2 2 + --- F(n) F(n + 1) + 3/154 F(n) F(n + 1) + --- F(n) F(n + 1) 154 308 351 2 4 3 5 - --- F(n) F(n + 1) - 3/22 F(n) F(n + 1) - 93/7 F(n) F(n + 1) 77 183 3 3 + --- F(n) F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 776 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 105 105 85 63 3 G(n) = - --- + --- F(n + 1) + --- F(n) - --- F(n) F(n + 1) 638 638 638 319 213 2 2 1337 2 3 123 3 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 319 638 319 1293 3 2 177 4 59 4 167 5 - ---- F(n) F(n + 1) + --- F(n) + -- F(n) F(n + 1) - --- F(n) 638 319 58 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 777 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 21471 2 21471 5 690 690 2 G(n) = ----- F(n) - ----- F(n) + ---- F(n + 1) - ---- F(n + 1) 12122 12122 6061 6061 40836 5 721 4 8565 5 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 6061 551 6061 25373 2 3 30792 2 4 44233 3 2 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 6061 6061 49713 3 3 64865 4 15417 4 2 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 12122 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 778 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 1418 1944 4 360 5 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 11 51 2 3 75 2 4 2070 3 2 - --- F(n) F(n + 1) + -- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 11 319 380 3 3 537 537 2 9095 2 - --- F(n) F(n + 1) - --- F(n) + --- F(n) + ---- F(n + 1) 11 638 638 638 864 5 155 6 1623 - --- F(n + 1) - --- F(n + 1) + ---- F(n + 1) 319 11 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 779 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 105 7 141 2 63 2 1519 6 G(n) = --- F(n + 1) - --- F(n) - --- F(n + 1) + ---- F(n) F(n + 1) 638 638 638 638 167 6 1029 2 4 351 4 2 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 638 638 638 401 4 3 10389 5 2 774 5 + --- F(n) F(n + 1) - ----- F(n) F(n + 1) - --- F(n) F(n + 1) 29 638 319 449 3 4 996 2 21 6 927 6 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n + 1) + --- F(n) 638 319 319 638 393 7 - --- F(n) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 780 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 615 615 10623 5067 3 G(n) = - ----- + ----- F(n + 1) + ----- F(n) - ---- F(n) F(n + 1) 12122 12122 12122 6061 9051 2 2 849 2 3 7059 3 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 12122 1102 6061 1047 3 2 6063 4 15 4 1461 5 + ---- F(n) F(n + 1) + ----- F(n) - -- F(n) F(n + 1) - ---- F(n) 1102 12122 29 1102 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 781 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 2 G(n) = 1/308 F(n) (-188 F(n + 1) + 246 F(n + 1) + 282 F(n) F(n + 1) 4 2 2 3 - 474 F(n) F(n + 1) - 376 F(n) F(n + 1) + 522 F(n) F(n + 1) 3 2 4 3 - 732 F(n) F(n + 1) + 720 F(n) F(n + 1) - 141 F(n) + 141 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 782 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1493154 7 1520631 3 2371437 3 514341 7 G(n) = -------- F(n) + ------- F(n + 1) + ------- F(n) - ------- F(n + 1) 1206139 4824556 1206139 4824556 2259711 3137994 4 456597 4 9665697 6 + ------- - ------- F(n) - ------ F(n + 1) - ------- F(n) F(n + 1) 1206139 1206139 219298 2412278 12134013 2 5 17625501 3 4 + -------- F(n) F(n + 1) - -------- F(n) F(n + 1) 4824556 2412278 5019606 3 2512764 3 208341 4 3 + ------- F(n) F(n + 1) - ------- F(n) F(n + 1) + ------ F(n) F(n + 1) 1206139 1206139 24244 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 783 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 47187 6 5520 5 25107 2 615 G(n) = ----- F(n) - ---- F(n) - ----- F(n) - ----- F(n + 1) 24244 6061 24244 12122 615 2 162 3 3 32679 4 + ----- F(n + 1) - --- F(n) F(n + 1) + ----- F(n) F(n + 1) 12122 551 12122 1203 4 2 6126 4 12741 5 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 24244 6061 12122 18123 2 3 831 2 4 20217 3 2 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 12122 1276 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 784 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 2 G(n) = 1/4620 F(n) (4306 F(n + 1) + 219 F(n + 1) - 13137 F(n) F(n + 1) 3 2 2 3 + 7310 F(n) F(n + 1) - 15270 F(n) F(n + 1) + 11615 F(n) F(n + 1) 3 3 4 2 5 - 43060 F(n) F(n + 1) + 49050 F(n) F(n + 1) + 2622 F(n) F(n + 1) 4 4 6 - 3655 F(n + 1) - 3980 F(n) + 3980 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 785 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 237921 2 8244 2 615 2 8244 3 G(n) = ------- F(n) F(n + 1) + ---- F(n) - ----- F(n + 1) - ---- F(n) 12122 6061 12122 6061 615 3 75 5 2 4 + ----- F(n + 1) + -- F(n) F(n + 1) - 15 F(n) F(n + 1) 12122 11 555 4 2 11490 6 1557 2 + --- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 22 551 209 27120 2 5 135 3 3 1755 3 4 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 551 11 1102 46485 4 3 4410 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 1102 551 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 786 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 405 6 34725 3 4 9681 4 G(n) = ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n + 1) 7562 3781 41591 11511 3 90615 3 11511 159933 2 + ----- F(n) + ----- F(n + 1) - ----- + ------ F(n) F(n + 1) 41591 83182 41591 83182 12554 3 71301 2 38193 2 2 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 41591 41591 41591 32640 2 5 51278 3 7905 7 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n + 1) 3781 41591 7562 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 787 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 2 G(n) = -3/40 F(n + 1) F(n) (-102 F(n + 1) F(n) - 238 F(n + 1) 3 3 5 2 + 582 F(n + 1) F(n) - 541 F(n + 1) F(n) - 14 F(n) + 79 F(n + 1) F(n) 6 + 234 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 788 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 18693 447 2 4 32 2 3 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 12122 6061 319 31344 5 8187 4 2 6337 4 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 6061 12122 1102 2562 3 3 6909 3 2 4551 2 17425 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) - ----- F(n) 551 1102 6061 12122 690 6 690 5 287 5 - ---- F(n + 1) + ---- F(n + 1) + --- F(n) 6061 6061 418 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 789 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 179 3 4 5549 5 2 40 5 G(n) = --- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n + 1) 33 924 231 751 2 5 751 7 257 40 - --- F(n) F(n + 1) + ---- F(n + 1) - --- F(n) + --- F(n + 1) 264 1848 924 231 200 2 3 2245 3 2 3541 6 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 231 924 924 1445 4 8783 4 3 751 3 257 3 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n + 1) + --- F(n) 924 1848 1848 924 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 790 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1387971 4 119550 7 217896 2 2 G(n) = ------- F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 83182 3781 41591 141639 2 3 53916 3 599499 4 - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 41591 41591 83182 89400 2 6 57711 142178 60300 8 + ----- F(n) F(n + 1) + ----- F(n) - ------ F(n + 1) - ----- F(n + 1) 3781 83182 41591 3781 810333 3 2 399225 3 5 57711 + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) - ----- 83182 7562 83182 176466 3 140348 5 + ------ F(n) F(n + 1) + ------ F(n + 1) 41591 41591 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 791 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4080 2 4 7755 2 5 60271 G(n) = ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 1102 6061 46485 7 57975 6 24320 3 3 - ----- F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 1102 551 319 117165 2 91215 3 4 44526 2 - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 1102 6061 9800 6 16990 2 23043 2 23043 3 - ---- F(n + 1) + ----- F(n + 1) + ----- F(n) - ----- F(n) 319 551 12122 12122 509955 3 22320 5 + ------ F(n + 1) + ----- F(n) F(n + 1) 12122 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 792 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 170 26325 8 78635 4 12163 2 2 G(n) = - --- + ----- F(n + 1) - ----- F(n + 1) - ----- F(n) F(n + 1) 231 154 462 462 2 4 63675 7 1835 + 5 F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 154 462 28125 3 5 32777 3 3 3 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - 30 F(n) F(n + 1) 77 462 596 3 75 2 6 5 + --- F(n) F(n + 1) - -- F(n) F(n + 1) + 55/2 F(n) F(n + 1) 77 22 145 2 170 2 145 6 + --- F(n + 1) + --- F(n) - --- F(n + 1) 12 231 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 793 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 1575 6 2 54750 5 3 11800 6 G(n) = ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 1102 6061 319 17547 2 18207 3 690 3 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n + 1) 12122 12122 6061 159825 3 690 33525 8 11800 5 2 - ------ F(n) - ---- + ----- F(n) - ----- F(n) F(n + 1) 12122 6061 12122 319 80 2 5 17925 2 6 77379 3 + --- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 12122 12122 15177 2 2 19699 2 88800 7 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 12122 6061 3360 7 + ---- F(n) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 794 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 529341 2 7 1708461 2 6 G(n) = ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 1140469 2280938 1091106 6 2 102960 4 4 + ------- F(n) F(n + 1) - ------ F(n) F(n + 1) 1140469 103679 10695636 5 4 6228825 6 3 - -------- F(n) F(n + 1) + ------- F(n) F(n + 1) 1140469 1140469 18870321 4 5 205920 5 3 + -------- F(n) F(n + 1) - ------ F(n) F(n + 1) 2280938 103679 2697417 3 5 3789990 3 6 + ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 1140469 1140469 273573 8 41184 7 2220912 7 2 + ------- F(n) F(n + 1) + ------ F(n) F(n + 1) - ------- F(n) F(n + 1) 2280938 103679 1140469 314838 8 185355 308826 5 185355 + ------- F(n) F(n + 1) + ------- F(n + 1) + ------- F(n) - ------- 1140469 2280938 1140469 2280938 20727 7 432297 8 - ------- F(n) F(n + 1) - ------- F(n) 1140469 2280938 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 795 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 4 2 2 G(n) = -3/308 F(n) (-50 F(n) F(n + 1) + 75 F(n) - 75 F(n) - 1379 F(n + 1) 2 2 2 4 3 + 200 F(n) F(n + 1) - 1090 F(n) F(n + 1) - 175 F(n) F(n + 1) 3 3 4 6 + 369 F(n) F(n + 1) + 3730 F(n) F(n + 1) + 25 F(n + 1) + 1320 F(n + 1) 5 - 2950 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 796 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2011887 3 96189 2 2 G(n) = -------- F(n) F(n + 1) + ----- F(n) F(n + 1) 2412278 63481 6140655 2 3 5454621 4 - ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 2412278 2412278 42326925 6 2 23965905 7 + -------- F(n) F(n + 1) + -------- F(n) F(n + 1) 2412278 2412278 242667 4 503145 910539 5 + ------- F(n) F(n + 1) - ------- F(n + 1) + ------- F(n) 1206139 2412278 1206139 503145 8 910539 8 51710265 5 3 + ------- F(n + 1) - ------- F(n) - -------- F(n) F(n + 1) 2412278 1206139 2412278 2974368 3 5 4822137 3 2 + ------- F(n) F(n + 1) + ------- F(n) F(n + 1) 1206139 1206139 10354464 3 - -------- F(n) F(n + 1) 1206139 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 797 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 2 G(n) = -3/40 F(n) F(n + 1) (12 F(n) - 1179 F(n + 1) + 463 F(n + 1) F(n) 7 6 2 + 1175 F(n + 1) - 2805 F(n + 1) F(n) - 126 F(n + 1) F(n) 5 2 4 3 - 170 F(n + 1) F(n) + 2630 F(n + 1) F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 798 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 2 G(n) = -3/308 F(n) (-17200 F(n) F(n + 1) - 646 F(n) F(n + 1) 2 3 3 4 7 + 770 F(n) F(n + 1) + 17550 F(n) F(n + 1) + 7350 F(n + 1) 2 3 5 + 2620 F(n) F(n + 1) - 7430 F(n + 1) - 440 F(n + 1) - 78 F(n) 3 3 2 4 + 78 F(n) - 1595 F(n) F(n + 1) + 935 F(n) F(n + 1) 2 5 - 2400 F(n) F(n + 1) + 486 F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 799 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 830373 5 18360 5 1788822 2 G(n) = ------ F(n + 1) - ----- F(n) F(n + 1) - ------- F(n) 7562 319 1206139 383625 9 864180 19710 3 3 - ------ F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 3781 109649 319 440775 2 7 863925 8 36261 2 3 + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 7562 3781 3781 7155 2 4 1979475 3 6 332349 4 - ---- F(n) F(n + 1) - ------- F(n) F(n + 1) - ------ F(n) F(n + 1) 638 7562 7562 104463 3 2 1788822 9816546 + ------ F(n) F(n + 1) + ------- F(n) - ------- F(n + 1) 3781 1206139 1206139 15975 6 30452310 2 + ----- F(n + 1) - -------- F(n + 1) 638 1206139 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 800 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(4 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 276 3 4 201 2 29331 9 459 7 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) - ----- F(n) - --- F(n) 77 88 616 308 201 7 1515 9 201 3 303 5 - --- F(n + 1) - ---- F(n + 1) + --- F(n + 1) + --- F(n + 1) 616 2464 616 176 30249 5 2727 11433 3 2 + ----- F(n) - ---- F(n + 1) + ----- F(n) F(n + 1) 616 2464 616 2109 4 3 2109 6 27975 6 3 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 154 616 1232 36555 7 2 23475 8 10323 4 5 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 616 224 2464 174 5 2 - --- F(n) F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 801 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 105 105 2 189 2 189 5 G(n) = ---- F(n + 1) + --- F(n + 1) + --- F(n) - --- F(n) 638 638 638 638 844 3 3 915 4 1050 4 2 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 638 319 32 5 669 2 3 1293 2 4 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 638 638 291 3 2 664 5 105 4 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 802 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(n + j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6477 7 19647 7 5609 2 4 G(n) = ----- F(n) + ----- F(n + 1) - ----- F(n) F(n + 1) 6061 24244 12122 176951 4 2 144259 2 21139 2 2037 3 + ------ F(n) F(n + 1) + ------ F(n) + ----- F(n + 1) - ---- F(n + 1) 12122 24244 24244 2204 207315 6 48765 4 3 5316 2 5 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 24244 12122 6061 93453 2 103914 5 19476 5 2 - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 24244 6061 6061 18379 6 6229 6 - ----- F(n + 1) - ---- F(n) 24244 1276 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 803 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(n + j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 219 3 669 2 105 6 1521 6 G(n) = ---- F(n) + ---- F(n) - --- F(n + 1) + ---- F(n) 319 3190 638 3190 453 4 2 13285 4 3 1098 5 + --- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 638 319 6895 5 2 2667 146 2 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 638 3190 29 2879 2 7485 2 5 2037 5 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 638 3190 105 6 105 7 + --- F(n) F(n + 1) + --- F(n + 1) 22 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 804 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 106 3 57 3 6444 2 G(n) = ---- F(n) F(n + 1) - -- F(n) - ---- F(n) F(n + 1) 11 58 319 75 2 6 57 64455 3 4 9225 3 5 + -- F(n) F(n + 1) + -- - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 22 58 638 22 903 3 73185 6 5275 7 - --- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 638 11 2073 2 342 2 2 1275 2 5 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 11 638 30075 3 15090 7 2175 8 2166 4 + ----- F(n + 1) - ----- F(n + 1) - ---- F(n + 1) + ---- F(n + 1) 638 319 11 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 805 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 58979 11490 6 3057 2 G(n) = ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 6061 551 418 27120 2 5 147 3 3 1755 3 4 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 551 11 1102 332 4 2 206 2 4 46485 4 3 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + ----- F(n) F(n + 1) 11 11 1102 21447 2 21447 3 237921 2 94 5 + ----- F(n) - ----- F(n) - ------ F(n) F(n + 1) + -- F(n) F(n + 1) 12122 12122 12122 11 615 2 615 3 - ----- F(n + 1) + ----- F(n + 1) 12122 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 806 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(n + 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 6 2 3 G(n) = -1/308 F(n) (201 F(n) + 5061 F(n) F(n + 1) + 2233 F(n) F(n + 1) 5 2 6 + 1249 F(n + 1) + 6720 F(n) F(n + 1) - 1503 F(n) F(n + 1) - 201 F(n) 3 2 2 5 3 - 4158 F(n) F(n + 1) - 7602 F(n) F(n + 1) + 609 F(n + 1) 2 7 4 - 2772 F(n) F(n + 1) - 915 F(n + 1) + 2079 F(n) F(n + 1) 5 - 1001 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 807 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(n + 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20541 3 5 293639 4 3 18207 7 G(n) = ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 12122 12122 12122 22041 2 34256 3 4 8244 6 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 6061 6061 2145 2 2 83526 4 4 172369 5 2 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 551 6061 6061 8211 5 3 47450 6 186141 6 2 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 638 6061 12122 84462 7 615 3 15936 4 615 - ----- F(n) F(n + 1) - ----- F(n + 1) + ----- F(n) + ----- 6061 12122 6061 12122 32487 7 - ----- F(n) 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 808 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 10525 2 1830 2 1830 5 126105 5 G(n) = ------ F(n) - ----- F(n + 1) + ----- F(n + 1) + ------ F(n) 4378 41591 41591 83182 359259 6 3 36935 6 1422684 5 4 - ------ F(n) F(n + 1) + ----- F(n) - ------- F(n) F(n + 1) 83182 41591 41591 4545309 4 5 12796 5 15421 3 3 + ------- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 83182 41591 41591 2049357 3 6 438483 4 213 4 2 - ------- F(n) F(n + 1) - ------ F(n) F(n + 1) - --- F(n) F(n + 1) 83182 83182 22 790695 3 2 151608 2 7 10749 4 + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 83182 41591 41591 411832 5 + ------ F(n) F(n + 1) 41591 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 809 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(n + 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 45960 4 3 17375 6 39410 2 5 G(n) = ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 319 690 3 34425 3 340573 2 + ---- F(n + 1) - ----- F(n) - ------ F(n) F(n + 1) 6061 12122 6061 8025 3 5 3255 4 194625 4 4 + ---- F(n) F(n + 1) + ---- F(n) + ------ F(n) F(n + 1) 551 1102 1102 48168 3 240525 2 6 7554 3 - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 551 1102 551 11290 3 4 49050 7 203101 2 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 319 551 12122 42879 2 2 690 + ----- F(n) F(n + 1) - ---- 1102 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 810 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(n + 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 G(n) = -1/4620 F(n) (28112 F(n) F(n + 1) - 3785 F(n) F(n + 1) 5 5 2 4 - 7570 F(n) F(n + 1) - 71355 F(n) F(n + 1) + 24385 F(n) - 6995 F(n) 6 3 5 5 3 - 17390 F(n) + 136360 F(n) F(n + 1) + 146338 F(n) F(n + 1) 6 2 7 4 - 119805 F(n) F(n + 1) + 49100 F(n) F(n + 1) + 4746 F(n + 1) 6 4 4 8 + 3785 F(n + 1) - 235450 F(n) F(n + 1) - 9401 F(n + 1) 4 2 + 78925 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 811 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(n + 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 56283 5 2421 4 690 9 939381 3 2 G(n) = ------ F(n) + ---- F(n) - ---- F(n + 1) - ------ F(n) F(n + 1) 12122 638 6061 12122 113850 4 5 34042 6 2 2214825 6 3 - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ------- F(n) F(n + 1) 551 319 12122 1027 3 18537 4 4150 4 4 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 12122 319 1115 3 5 567075 3 6 289875 4 + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 638 6061 12122 967 7 5865 5 4 46615 5 3 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 638 638 9243 3 690 4452 - ---- F(n) F(n + 1) + ---- + ---- F(n) 319 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 812 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 15 2 2202 6 4095 5 2 G(n) = --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 58 319 638 1281 2 5 250 5 3 105 3 489 3 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n + 1) - --- F(n) 638 11 638 319 18 3 63 6 7 105 + -- F(n) F(n + 1) - --- F(n) F(n + 1) + 2/55 F(n) F(n + 1) + --- 55 638 638 216 2 2 4 4 499 6 2 + --- F(n) F(n + 1) + 37/2 F(n) F(n + 1) + --- F(n) F(n + 1) 55 55 78 7 7 15 8 - -- F(n) F(n + 1) + 3/638 F(n) + -- F(n) 11 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 813 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(2 n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 35295 3 5 10702 4 58191 2 3 G(n) = ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 1276 319 1595 20673 4 1183 9 26745 8 15034 - ----- F(n) - ---- F(n) + ----- F(n) + ----- F(n) 2552 66 2552 957 9525 7 105 5775 2 6 34101 2 2 - ---- F(n) F(n + 1) - --- + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 1276 638 232 2552 94907 8 9021 3 97415 8 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 1914 1276 1914 398579 7 2 622327 3 2 11133 3 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 3190 9570 1276 468947 2 7 105 9 + ------ F(n) F(n + 1) + --- F(n + 1) 4785 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 814 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1766993 1953129 4 48975 5 1559 4 G(n) = ------- - ------- F(n) F(n + 1) - ----- F(n) - ---- F(n) 12122 6061 12122 11 2780 2 2 412575 8 7 + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) + 1630 F(n) F(n + 1) 11 1102 93225 3 6 3 5 42075 4 5 + ----- F(n) F(n + 1) - 1410 F(n) F(n + 1) + ----- F(n) F(n + 1) 1102 58 1555671 3 2 3524 3 1027125 2 7 - ------- F(n) F(n + 1) - ---- F(n) F(n + 1) - ------- F(n) F(n + 1) 12122 11 1102 113850 2 3 8 2 6 + ------ F(n) F(n + 1) - 670 F(n + 1) - 15 F(n) F(n + 1) 551 5766 4 153243 305871 5 + ---- F(n + 1) + ------ F(n + 1) - ------ F(n + 1) 11 6061 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 815 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 24712 5 279585 8 605995 9 G(n) = ------ F(n) F(n + 1) + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 11 638 319 994001 2 4 686445 2 7 1020 2 + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) + ---- F(n) 638 638 319 110538 164277 4 223521 2 3 + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 319 319 638 1487390 2 8 10170 3 6 55930 3 7 - ------- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 319 2895 4 365049 4 2 286035 4 5 - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 22 638 319 1240165 4 6 121347 5 24579 + ------- F(n) F(n + 1) - ------ F(n) F(n + 1) + ----- F(n) 319 319 319 25599 5 105 9 105 10 - ----- F(n) - --- F(n + 1) + --- F(n + 1) 319 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 816 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 36916 8 12 5944 4 G(n) = -1 - ----- F(n + 1) + 8575/3 F(n + 1) - ---- F(n + 1) 15 15 103 3 2 10 2 6 + --- F(n) F(n + 1) + 850 F(n) F(n + 1) - 4424/5 F(n) F(n + 1) 15 2 2 11 8624 7 - 141/5 F(n) F(n + 1) - 7200 F(n) F(n + 1) + ---- F(n) F(n + 1) 15 3 9 18032 3 5 3 + 16100/3 F(n) F(n + 1) + ----- F(n) F(n + 1) + 562/5 F(n) F(n + 1) 15 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 817 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 70101055 292162181 11 27075883 2 2 G(n) = -------- + --------- F(n) F(n + 1) + -------- F(n) F(n + 1) 624283 624283 65714 879 2 3 1512257777 2 6 2967 3 2 - --- F(n) F(n + 1) + ---------- F(n) F(n + 1) - ---- F(n) F(n + 1) 551 1248566 1102 349196484 3 60807521 10 2 + --------- F(n) F(n + 1) - -------- F(n) F(n + 1) 624283 43054 422478664 11 2240 4 - --------- F(n) F(n + 1) + ---- F(n) F(n + 1) 624283 551 418015161 7 36785280 9 3 + --------- F(n) F(n + 1) - -------- F(n) F(n + 1) 624283 32857 136214431 8 69152940 12 3144 15757 - --------- F(n + 1) - -------- F(n) - ---- F(n) + ----- F(n + 1) 624283 624283 6061 12122 323611829 12 456059001 4 1627 5 - --------- F(n + 1) + --------- F(n + 1) - ---- F(n + 1) 1248566 1248566 1102 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 818 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 6415 4 207 8 2 9 G(n) = ---- F(n + 1) + --- F(n) F(n + 1) + 3/7 F(n) F(n + 1) 28 70 831 3 7 3 9 1821 4 2 - --- F(n) F(n + 1) - 1875/2 F(n) F(n + 1) - ---- F(n) F(n + 1) 280 560 172315 4 4 69 2 8125 6 6 - ------ F(n) F(n + 1) + --- F(n + 1) + ---- F(n) F(n + 1) 168 112 28 7 6375 2 10 1259 3 3 - 595/2 F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 14 280 135 7 3 2 129 6 4 + --- F(n) F(n + 1) - 9/28 F(n) + --- F(n) F(n + 1) 28 112 7 5 3713 3 5 3189 3 + 6375/7 F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 14 14 2755 57 6 45 10 26273 8 4 - ---- - -- F(n + 1) + --- F(n + 1) - ----- F(n) + 539/2 F(n) 24 56 112 168 19205 8 - ----- F(n + 1) 168 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 819 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 442800 8 13725 2 6 24786 3 3125 G(n) = ------- F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- 6061 12122 6061 12122 20 5 8997 9950 186444 3 + -- F(n + 1) + ----- F(n) - ---- F(n + 1) - ------ F(n) F(n + 1) 11 12122 6061 6061 1077525 7 73809 2 2 36 3 2 + ------- F(n) F(n + 1) + ----- F(n) F(n + 1) + -- F(n) F(n + 1) 6061 6061 11 2 3 101 4 880335 4 - 1/22 F(n) F(n + 1) - --- F(n) F(n + 1) + ------ F(n + 1) 22 12122 929475 3 5 - ------ F(n) F(n + 1) 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 820 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 5 53264 4 G(n) = 13/3 F(n + 1) - 6/11 F(n) + 26/3 F(n) F(n + 1) - ----- F(n + 1) 231 53600 8 6 400 2 6 + ----- F(n + 1) - 13/3 F(n + 1) + --- F(n) F(n + 1) 231 77 3 3 16 37600 3 5 - 43/3 F(n) F(n + 1) - -- + ----- F(n) F(n + 1) 11 77 130400 7 2976 2 2 2 4 - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) + 5 F(n) F(n + 1) 231 77 7552 3 3008 3 - 8/33 F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 77 231 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 821 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1519371 4 764100 8 419 7 1586069 3 G(n) = ------- F(n + 1) - ------ F(n + 1) + --- F(n + 1) - ------- F(n + 1) 7562 3781 11 41591 148767 3 97875 7 293669 2 + ------ F(n) + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 83182 199 41591 127413 2 2 61 2 5 1601775 3 5 + ------ F(n) F(n + 1) + -- F(n) F(n + 1) - ------- F(n) F(n + 1) 3781 22 3781 321084 3 1441013 2 18225 2 6 - ------ F(n) F(n + 1) + ------- F(n) F(n + 1) - ----- F(n) F(n + 1) 3781 83182 7562 42714 3 100779 1669 3 4 - ----- F(n) F(n + 1) + ------ + ---- F(n) F(n + 1) 3781 83182 22 2067 6 - ---- F(n) F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 822 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 11345 9 114490 8 26450 8 G(n) = ------ F(n) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 12122 6061 6061 5669 4 2 1692 2 12899 2 - ----- F(n) F(n + 1) + ---- F(n + 1) - ----- F(n) 12122 6061 12122 10273 2 4 460197 2 7 234007 3 2 + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 6061 12122 6061 54 3 3 21347 5 393100 6 3 - -- F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 11 6061 6061 69447 7 2 65503 2 3 622 6 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n + 1) 6061 6061 6061 112177 114317 5 + ------ F(n + 1) - ------ F(n + 1) 12122 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 823 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 4 3 5 G(n) = 1/231 F(n) (105612 F(n) F(n + 1) - 15326 F(n) F(n + 1) 2 6 2 4 7 - 101537 F(n) F(n + 1) - 1808 F(n) F(n + 1) + 43826 F(n) F(n + 1) 5 8 + 157 F(n) F(n + 1) - 37 F(n) F(n + 1) + 112 F(n + 1) 3 3 2 4 2 + 2769 F(n) F(n + 1) + 208 F(n) - 1734 F(n) F(n + 1) 3 2 2 3 - 3978 F(n) F(n + 1) + 15565 F(n) F(n + 1) - 44274 F(n) F(n + 1) 4 6 + 485 F(n) - 40 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 824 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 659 4 3 1070 10 10948 3 G(n) = ---- F(n) F(n + 1) - ---- F(n + 1) + ----- F(n) 22 6061 6061 79395 2 4050429 2 4 416 2 5 - ----- F(n) F(n + 1) + ------- F(n) F(n + 1) + --- F(n) F(n + 1) 12122 12122 11 23732185 2 8 967451 3 3 101 3 4 - -------- F(n) F(n + 1) - ------ F(n) F(n + 1) - --- F(n) F(n + 1) 12122 6061 22 91918 2 9889380 4 6 249782 5 + ----- F(n) F(n + 1) + ------- F(n) F(n + 1) - ------ F(n) F(n + 1) 6061 6061 319 169 6 4837165 9 7235 2 - --- F(n) F(n + 1) + ------- F(n) F(n + 1) + ---- F(n) 11 6061 6061 438660 3 7 86897 1070 3 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ---- F(n + 1) 6061 6061 6061 87047 4 2 + ----- F(n) F(n + 1) 1102 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 825 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 9 G(n) = -1/84 F(n) (78 F(n) + 6 F(n) + 78750 F(n) F(n + 1) 2 5 6 3 8 - 91710 F(n) F(n + 1) + 109695 F(n) F(n + 1) + 643125 F(n) F(n + 1) 3 4 3 2 3 + 89385 F(n) F(n + 1) + 33 F(n) F(n + 1) - 25616 F(n + 1) 2 3 2 10 + 108 F(n) F(n + 1) - 964 F(n) F(n + 1) - 834375 F(n) F(n + 1) 2 11 7 + 5934 F(n) F(n + 1) + 333750 F(n + 1) - 308130 F(n + 1) 5 4 + 30 F(n + 1) - 36 F(n + 1) - 63 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 826 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(4 n - 2 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 192243 4 7 470 4 4 37171 4 3 G(n) = ------ F(n) F(n + 1) + --- F(n) F(n + 1) + ----- F(n) F(n + 1) 12760 11 6380 1532 3 9 10565 3 8 41597 8 3 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 638 2552 47208 11 223 10 88116 9 3 + ----- F(n) F(n + 1) - --- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 319 372 11 2935 3 1955 11 12494 4 + --- F(n) + ---- F(n + 1) + ---- F(n + 1) - ----- F(n) 319 2552 2552 319 12954 12 80 8 215 7 254 4 194 + ----- F(n) - --- F(n + 1) - --- F(n + 1) + --- F(n + 1) - --- 319 319 116 319 319 25293 8 4 40296 10 2 412 5 2 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 55 1809 4 8 22908 9 2 37872 5 3 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 1595 319 10477 5 6 + ----- F(n) F(n + 1) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 827 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 80875 8 45 3 2 59 4 G(n) = ------ F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 638 22 22 40196 4 443 2264 3 650 2 6 + ----- F(n + 1) + --- - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 319 638 319 319 6740 2 2 84900 3 5 17032 3 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 319 98400 7 13 2 3 25 5 195 + ----- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n + 1) + --- F(n) 319 22 22 638 685 - --- F(n + 1) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 828 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(4 n - 2 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 4 2 5 G(n) = -1/84 F(n) (16750 F(n) F(n + 1) - 1750 F(n) F(n + 1) 2 3 6 2 + 420 F(n) F(n + 1) - 17125 F(n) F(n + 1) + 2685 F(n) F(n + 1) 7 3 3 2 + 7250 F(n + 1) - 7290 F(n + 1) - 504 F(n) F(n + 1) 2 4 - 730 F(n) F(n + 1) + 168 F(n) F(n + 1) - 21 F(n) + 126 F(n + 1) 5 3 - 84 F(n + 1) + 105 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 829 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(4 n - 2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6495 3 1419 4 11825 3 5 1070 G(n) = ----- F(n) + ---- F(n) - ----- F(n) F(n + 1) + ---- 12122 1102 551 6061 156537 2 1070 3 26814 2 + ------ F(n) F(n + 1) - ---- F(n + 1) - ----- F(n) F(n + 1) 12122 6061 6061 52855 3 289 6 53075 7 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) + ----- F(n) F(n + 1) 551 22 551 336 4 3 122375 4 4 57 3 4 - --- F(n) F(n + 1) + ------ F(n) F(n + 1) + -- F(n) F(n + 1) 11 551 22 187 3 247775 2 6 631 2 5 - --- F(n) F(n + 1) - ------ F(n) F(n + 1) + --- F(n) F(n + 1) 19 1102 22 39391 2 2 + ----- F(n) F(n + 1) 1102 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 830 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(4 n - 2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 116043 5 301 2 265 5 1145 3 2 G(n) = ------ F(n + 1) - --- F(n) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 22 29 2 4 24462 2 3 582625 8 - 3/11 F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 319 638 99421 4 480025 3 6 106 3 3 - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) + --- F(n) F(n + 1) 638 638 11 12900 2 7 54 6 618 - ----- F(n) F(n + 1) + -- F(n + 1) + --- F(n) F(n + 1) 319 11 319 236725 9 337 4599 1546 2 - ------ F(n + 1) - --- F(n) + ---- F(n + 1) - ---- F(n + 1) 638 638 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 831 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(4 n - 2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 5 2 4 G(n) = 1/84 F(n) (152 F(n) + 51 F(n) F(n + 1) - 606 F(n) F(n + 1) 2 4 2 3 5 - 33 F(n) - 486 F(n) F(n + 1) + 49 - 5450 F(n) F(n + 1) 3 3 7 2 2 + 705 F(n) F(n + 1) + 16325 F(n) F(n + 1) + 5848 F(n) F(n + 1) 2 4 4 5 - 51 F(n + 1) + 38875 F(n) F(n + 1) + 219 F(n) F(n + 1) 2 6 3 3 - 37700 F(n) F(n + 1) - 1573 F(n) F(n + 1) - 16325 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 832 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(4 n - 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 6 1315 3 5 34975 4 5 G(n) = 355/2 F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 22 551 3645 4 4 232700 5 4 12937 5 - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) - ----- F(n) 22 551 6061 141 12531 4 25739 4 929 5 + ---- F(n + 1) + ----- F(n) + ----- F(n + 1) + ---- F(n + 1) 6061 418 418 6061 340 8 373891 23653 2 2 2153431 2 3 - --- F(n + 1) - ------ - ----- F(n) F(n + 1) + ------- F(n) F(n + 1) 11 12122 418 12122 190975 2 7 14486 3 33998 3 2 - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 1102 209 551 224875 3 6 187479 4 + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 551 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 833 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(4 n - 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 169 2 248 3 237 3 390 6 129 6 G(n) = ---- F(n + 1) + --- F(n) + ---- F(n + 1) + --- F(n) + --- F(n + 1) 638 319 3190 319 638 723 2 2333 3 4 37 7 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n + 1) 1595 638 3190 208433 4 6 47121 3 3 46042 3 7 - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 638 319 319 1483 4 2 26 4 3 3899 5 + ---- F(n) F(n + 1) + -- F(n) F(n + 1) - ---- F(n) F(n + 1) 29 11 319 8111 5 2 22842 5 5 1164 6 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 3190 319 319 231633 6 4 179 2 8 + ------ F(n) F(n + 1) + --- F(n) F(n + 1) 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 834 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(4 n - 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 2 7 G(n) = 1/420 F(n) (8370 F(n) F(n + 1) + 396 F(n) F(n + 1) - 4158 F(n + 1) 8 7 2 6 3 + 21415 F(n) F(n + 1) - 9520 F(n) F(n + 1) + 24460 F(n) F(n + 1) 6 5 4 4 5 - 1410 F(n) F(n + 1) - 82700 F(n) F(n + 1) + 49115 F(n) F(n + 1) 5 2 5 3 + 4284 F(n) F(n + 1) - 2039 F(n + 1) + 3014 F(n + 1) + 3858 F(n + 1) 4 3 7 5 - 13140 F(n) F(n + 1) + 270 F(n) + 1903 F(n) - 3433 F(n) 9 - 685 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 835 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(4 n - 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 197 11 101 8 61355 7 4 3743 3 G(n) = ----- F(n + 1) + ---- F(n) + ----- F(n) F(n + 1) - ---- F(n + 1) 1276 1276 638 6380 1003 4 696943 10 193155 9 2 + ---- F(n + 1) - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 638 6380 638 6051 7 599 6 2 4827 5 3 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 319 638 499 4 4 238793 8 3 34143 4 3 - ---- F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 1276 1276 1276 1887 3 43 2 804063 6 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) + ------ F(n) F(n + 1) 638 290 6380 30271 5 6 1695 1082 7 21 8 - ----- F(n) F(n + 1) - ---- + ---- F(n + 1) - --- F(n + 1) 1595 1276 1595 116 1117 11 - ---- F(n) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 836 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 5 7 G(n) = -1/60 F(n) (56 F(n + 1) + 252 F(n) F(n + 1) + 60 F(n) + 105 F(n) 5 6 4 3 7 + 140 F(n + 1) - 520 F(n) F(n + 1) - 1960 F(n) F(n + 1) - 96 F(n + 1) 4 3 4 4 - 240 F(n) F(n + 1) + 1190 F(n) F(n + 1) + 320 F(n) F(n + 1) 5 2 + 868 F(n) F(n + 1) - 130 F(n + 1) - 45 F(n)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 837 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(4 n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 9975 4 4 23674 3 23800 7 G(n) = ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 58 319 319 255 2 8834 2 2 55475 2 6 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 58 319 319 84 3 57 3 4 303 3 20 3 - -- F(n) F(n + 1) + -- F(n) F(n + 1) + --- F(n) - --- F(n + 1) 11 22 319 319 5250 3 5 8163 2 20 315 4 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- + --- F(n) 319 638 319 319 336 4 3 631 2 5 289 6 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 11 22 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 838 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(4 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2268 4 37800 3 5 1716689 2 3 G(n) = ---- F(n) - ----- F(n) F(n + 1) - ------- F(n) F(n + 1) 551 551 12122 22125 4 5 722925 4 4 1000 3 6 - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 29 1102 29 15750 3 24225 2 7 369600 2 6 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 551 29 551 10075 8 156450 7 2114743 4 - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + ------- F(n) F(n + 1) 29 551 6061 157332 3 690 12108 5 573505 3 2 - ------ F(n) F(n + 1) - ---- + ----- F(n) + ------ F(n) F(n + 1) 551 6061 6061 12122 142993 4 57939 2 2 690 - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n + 1) 12122 551 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 839 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(4 n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 7 25 9 5 4 G(n) = 1/8 F(n + 1) - 1/12 F(n) + -- F(n) + 247/6 F(n) F(n + 1) 12 2 7 5 2 - 15/8 F(n) F(n + 1) - 3/8 F(n + 1) - 20/3 F(n) F(n + 1) 7 179 3 6 25 6 - 1/8 F(n + 1) + --- F(n) F(n + 1) + -- F(n) F(n + 1) 12 12 6 3 199 4 3 4 5 - 181/4 F(n) F(n + 1) + --- F(n) F(n + 1) - 119/4 F(n) F(n + 1) 24 8 7 2 2 5 - 91/8 F(n) F(n + 1) + 193/6 F(n) F(n + 1) + 7/8 F(n) F(n + 1) 79 3 4 5 - -- F(n) F(n + 1) + 3/8 F(n + 1) 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 840 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(4 n - j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 629 8 20 9 647 9 20 8 G(n) = ---- F(n) + --- F(n + 1) - --- F(n) - --- F(n + 1) 319 319 319 319 1983 7 36019 8 6987 2 3 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 1595 3190 3190 5341 3 18582 4 28263 5 4 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 3190 1595 638 28593 6 2 4613 6 3 9203 7 2 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 3190 110 319 4137 8 1854 2 6 17785 2 7 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 638 2479 7 129 5 3 5283 2 2 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 22 3190 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 841 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(4 n - j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 9 G(n) = -1/2460 F(n) (10612 F(n + 1) + 36900 F(n + 1) + 63996 F(n) 7 3 3 - 19680 F(n) - 54566 F(n) + 20090 F(n) - 36080 F(n + 1) 6 3 2 5 2 - 711952 F(n) F(n + 1) + 113160 F(n) F(n + 1) - 29930 F(n) F(n + 1) 8 6 4 + 236665 F(n) F(n + 1) - 123000 F(n) F(n + 1) + 86766 F(n) F(n + 1) 2 9 8 + 47970 F(n) F(n + 1) - 121258 F(n + 1) + 151482 F(n) F(n + 1) 7 2 5 + 220229 F(n) F(n + 1) + 108596 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 842 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(4 n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 8 7475 2 7 3 7 G(n) = -775 F(n) F(n + 1) + ---- F(n) F(n + 1) - 7600 F(n) F(n + 1) 11 250 3 6 241232 3 3 1024 3 2 - --- F(n) F(n + 1) - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 11 319 11 6125 8 474849 5 5117 4 - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 22 319 22 313987 2 4 3075 2 3 10 + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) - 7775/2 F(n + 1) 319 22 982 2 60139 2 9 10750 + --- F(n) + ----- F(n + 1) + 9675 F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 319 12675 4 5 2359987 6 5 6719 - ----- F(n) F(n + 1) + ------- F(n + 1) + 21 F(n + 1) - ---- F(n + 1) 22 638 319 932 + --- F(n) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 843 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(4 n - j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 4 2 G(n) = 1/120 F(n) (-425 + 1530 F(n) F(n + 1) - 18182 F(n) F(n + 1) 3 5 9 8 2 + 3750 F(n) F(n + 1) - 4820 F(n) F(n + 1) + 33007 F(n) F(n + 1) 6 2 8 6 4 - 4875 F(n) F(n + 1) - 375 F(n + 1) - 55 F(n + 1) + 750 F(n + 1) 2 2 8 5 5 + 165 F(n + 1) + 2435 F(n) F(n + 1) - 56550 F(n) F(n + 1) 5 3 4 6 4 4 + 4500 F(n) F(n + 1) + 53190 F(n) F(n + 1) - 6000 F(n) F(n + 1) 3 7 4 10 3 3 - 17084 F(n) F(n + 1) + 435 F(n) + 710 F(n) + 7894 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 844 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 20 347 324 3 686 2 2 G(n) = --- - --- F(n + 1) + --- F(n) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 319 319 12 2 3 848 3 77 3 2 293 4 + -- F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) F(n + 1) + --- F(n) 29 319 29 319 72 4 31 5 - -- F(n) F(n + 1) - -- F(n) 29 29 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 845 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 5 3 3 G(n) = - 1/14 + 174/7 F(n + 1) - 405/7 F(n) F(n + 1) + 58 F(n) F(n + 1) 4 2 27 2 + 1/14 F(n + 1) - 174/7 F(n + 1) - -- F(n) + 60/7 F(n) F(n + 1) 14 3 2 2 2 4 - 1/21 F(n) F(n + 1) - 3/14 F(n) F(n + 1) - 48/7 F(n) F(n + 1) 3 + 1/3 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 846 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 387253 7 6261 4 6048 3 1452 2 2 G(n) = ------ F(n) - ---- F(n) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 24244 6061 6061 551 18996 3 79 5 2 167587 6 + ----- F(n) F(n + 1) - -- F(n) F(n + 1) + ------ F(n) F(n + 1) 6061 19 12122 286717 3 690 3 690 42871 6 - ------ F(n) + ---- F(n + 1) - ---- + ----- F(n) F(n + 1) 24244 6061 6061 24244 70047 2 159487 4 3 - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 24244 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 847 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2216 3 2 3149 3 3 1763 4 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 638 319 1715 4 2 195 5 75 4 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) - -- F(n) F(n + 1) 638 638 58 871 5 1244 2 3 1981 2 4 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 319 638 1191 5 85 6 20 20 2 - ---- F(n) - --- F(n) + --- F(n + 1) - --- F(n + 1) 638 638 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 848 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 452 7 5 G(n) = -1/21 F(n) + 5/42 F(n + 1) + --- F(n + 1) - 5/42 F(n + 1) 21 1789 6 583 3 4 2041 5 2 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 42 42 42 64 3 4 3 2 + -- F(n) + 5/14 F(n) F(n + 1) - 5/21 F(n) F(n + 1) 21 673 2 2 3 2 5 - --- F(n) F(n + 1) - 5/42 F(n) F(n + 1) - 48/7 F(n) F(n + 1) 42 452 3 - --- F(n + 1) 21 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 849 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3940 7 20 7 7977 3 16543 6 G(n) = ----- F(n) - --- F(n + 1) + ---- F(n) - ----- F(n) F(n + 1) 319 319 638 638 5287 4 2 7191 4 3 3 3 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - 14 F(n) F(n + 1) 319 638 3781 2 4 69 2 5 564 2 + ---- F(n) F(n + 1) - -- F(n) F(n + 1) + --- F(n) F(n + 1) 638 58 319 3469 5 4833 5 2 541 5 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 319 20 6 1817 6 + --- F(n + 1) + ---- F(n) 319 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 850 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 103 2 4 3 19 3 G(n) = ---- F(n) F(n + 1) - 19/6 F(n) F(n + 1) + -- F(n) F(n + 1) 14 42 57 2 2 19 47 2 + 60/7 F(n) F(n + 1) + -- F(n) F(n + 1) + -- - -- F(n) 28 28 28 2 19 4 6 3 3 - 167/7 F(n + 1) - -- F(n + 1) + 167/7 F(n + 1) + 113/2 F(n) F(n + 1) 28 775 5 - --- F(n) F(n + 1) 14 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 851 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1070 3 15786 3 1341 6 G(n) = ---- F(n + 1) + ----- F(n) - ---- F(n) F(n + 1) 6061 6061 38 3239 2 5 2943 4 3 108 3 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 38 38 6061 76507 2 958 2 2 10484 3 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 551 6061 54 3 4 2594 4 425857 2 1070 + -- F(n) F(n + 1) - ---- F(n) + ------ F(n) F(n + 1) - ---- 19 6061 12122 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 852 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 1583 182 4 439 5 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 638 29 29 1499 2 3 110 2 4 7463 3 2 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 29 638 527 3 3 1118 2242 2 1098 5 + --- F(n) F(n + 1) + ---- F(n + 1) - ---- F(n + 1) - ---- F(n + 1) 29 319 319 319 202 6 379 259 2 + --- F(n + 1) - --- F(n) - --- F(n) 29 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 853 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 2 4 G(n) = -1/84 F(n) (-228 F(n) + 290 F(n) F(n + 1) + 85 F(n + 1) 3 2 2 4 - 170 F(n) F(n + 1) - 3963 F(n + 1) - 1758 F(n) F(n + 1) 3 5 - 205 F(n) F(n + 1) + 1269 F(n) F(n + 1) - 8934 F(n) F(n + 1) 3 3 6 4 + 9678 F(n) F(n + 1) + 3876 F(n + 1) + 60 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 854 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 14011 3237 34537 5 1150 2 6 G(n) = ----- F(n) - ---- F(n + 1) + ----- F(n + 1) + ---- F(n) F(n + 1) 12122 551 6061 19 192198 3 6320 8 15690 7 + ------ F(n) F(n + 1) + ---- F(n + 1) - ----- F(n) F(n + 1) 6061 19 19 921050 3 24915 3 5 38906 2 3 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 38 6061 443103 2 2 151251 4 203397 3 2 - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 6061 12122 12122 50377 3979643 4 - ----- - ------- F(n + 1) 12122 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 855 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 2 271 7 6 459 2 20 3 G(n) = --- F(n + 1) + --- F(n) + 3/22 F(n) + --- F(n) - --- F(n + 1) 319 638 319 319 63 5 3099 2 10467 2 - -- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 11 3190 3190 327 5 2 2968 6 1384 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 29 1595 319 15127 6 225 2 4 2702 2 5 + ----- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 3190 22 319 135 5 - --- F(n) F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 856 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 47 2 93 4 2 47 2 4 145 8 G(n) = -- F(n + 1) + -- F(n) F(n + 1) + -- F(n) F(n + 1) + --- F(n + 1) 84 14 14 168 787 8 241 2 2 109 3 3 - --- F(n) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 168 42 14 4 4 983 5 3 1117 6 2 - 207/8 F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 21 42 39 6 47 6 97 4 5 + -- F(n) - -- F(n + 1) - -- F(n + 1) - 5 F(n) F(n + 1) 28 84 84 367 7 + --- F(n) F(n + 1) + 7/24 21 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 857 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4950 8 2775 7 120 2 6 G(n) = ---- F(n + 1) - ---- F(n + 1) - --- F(n) F(n + 1) 29 22 29 5275 3 6225 3 4 80515 3 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n + 1) 638 22 638 54205 4 265 10630 3 5 45201 3 - ----- F(n + 1) - --- + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 29 638 11955 7 9103 2 8350 2 2 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 29 638 319 375 2 5 692 3 31455 2 + --- F(n) F(n + 1) - --- F(n) - ----- F(n) F(n + 1) 22 319 638 6 + 300 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 858 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 5 2 G(n) = -1/15 F(n) (-294 F(n) F(n + 1) + 1068 F(n) F(n + 1) 2 3 4 6 - 328 F(n) F(n + 1) + 5904 F(n) F(n + 1) - 6402 F(n) F(n + 1) 3 7 3 - 2681 F(n + 1) + 2673 F(n + 1) + 60 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 859 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3619835 7 2 1043004 8 16635 3 G(n) = ------- F(n) F(n + 1) - ------- F(n) F(n + 1) - ----- F(n) F(n + 1) 83182 41591 41591 138739 3 2 43216 5 4 3751791 6 3 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ------- F(n) F(n + 1) 83182 2189 83182 3327 7 21343 4 71529 3 5 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 41591 41591 41591 49905 5 3 1512 6 2 15069 2 7 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 41591 3781 83182 47406 2 6 45611 3 6 756 7 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 41591 41591 3781 1830 10812 8 1830 9 236904 9 + ----- + ----- F(n) - ----- F(n + 1) + ------ F(n) 41591 41591 41591 41591 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 860 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 6 3 G(n) = 1/84 F(n) (-8 F(n + 1) + 52 F(n + 1) + 16 F(n) F(n + 1) 3 5 4 2 + 114 F(n) F(n + 1) - 792 F(n) F(n + 1) + 1239 F(n) F(n + 1) 5 4 3 3 - 156 F(n) F(n + 1) - 53 F(n) - 1030 F(n) F(n + 1) 2 2 2 4 6 - 122 F(n) F(n + 1) + 519 F(n) F(n + 1) + 221 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 861 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 97117 2669 8 22683 4 69 23158 G(n) = - ----- + ----- F(n + 1) + ----- F(n + 1) - -- F(n) + ----- F(n + 1) 48488 22040 11020 55 30305 107523 5 28508 5 11093 8 12646 3 + ------ F(n) - ----- F(n + 1) - ----- F(n) - ----- F(n) F(n + 1) 60610 30305 4408 2755 11929 5 3 20012 4 1934 2 3 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 2755 6061 551 869 4 4 32441 6 2 24353 7 + --- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 232 2204 1102 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 862 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2938 147596 19778 2 7 G(n) = ----- F(n) + ------ F(n + 1) - ----- F(n) F(n + 1) 71 1065 71 12039 2 3 7203 9 3364 9 + ----- F(n) F(n + 1) - ---- F(n + 1) + ---- F(n) 71 71 71 23821 8 39721 4 89245 3 2 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 71 213 213 39551 5 - ----- F(n + 1) 1065 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 863 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 3 7 G(n) = -1/168 F(n) (60 F(n + 1) - 12 F(n + 1) - 238 F(n + 1) + 102 F(n + 1) 5 7 4 3 4 - 94 F(n) + 766 F(n) - 7735 F(n) F(n + 1) + 360 F(n) F(n + 1) 2 2 5 3 2 - 2121 F(n) F(n + 1) + 203 F(n) F(n + 1) - 840 F(n) F(n + 1) 5 2 6 2 3 + 9744 F(n) F(n + 1) - 815 F(n) F(n + 1) + 620 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 864 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 38645 2 1070 2 145991 6 3 G(n) = ----- F(n) - ---- F(n + 1) - ------ F(n) F(n + 1) 6061 6061 2204 348739 7 2 17275 8 588589 4 5 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 12122 6061 24244 145457 2 3 90 2 4 5327 8 + ------ F(n) F(n + 1) - -- F(n) F(n + 1) - ---- F(n) F(n + 1) 24244 11 6061 2118 38371 2 7 93296 5 272949 9 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) + ------ F(n) 6061 24244 6061 12122 1070 9 6 200 3 3 105 5 + ---- F(n + 1) - 15/2 F(n) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 6061 11 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 865 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4925 2 7 1625 3 4 6 G(n) = ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + 575/4 F(n) F(n + 1) 14 12 725 7 9 185279 5 151 - --- F(n + 1) + 15875/7 F(n + 1) - ------ F(n + 1) + --- F(n) 12 84 21 5221 725 3 25 3 565 2 - ---- F(n + 1) + --- F(n + 1) - -- F(n) + --- F(n) F(n + 1) 84 12 21 84 43331 2 3 2 5 25589 3 2 - ----- F(n) F(n + 1) + 25/3 F(n) F(n + 1) + ----- F(n) F(n + 1) 84 84 63275 3 6 8 4 + ----- F(n) F(n + 1) - 11225/2 F(n) F(n + 1) + 6624/7 F(n) F(n + 1) 14 655 2 - --- F(n) F(n + 1) 28 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 866 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 9 9 G(n) = -1/396 F(n) (981 F(n) - 177 F(n) + 2760 F(n) - 218 F(n + 1) 2 2 3 2 5 - 354 F(n) F(n + 1) + 1452 F(n) F(n + 1) + 472 F(n) F(n + 1) 3 2 3 4 4 3 + 8184 F(n) F(n + 1) - 1121 F(n) F(n + 1) - 118 F(n) F(n + 1) 4 5 5 2 5 4 - 12420 F(n) F(n + 1) + 1121 F(n) F(n + 1) + 54309 F(n) F(n + 1) 6 6 3 7 2 + 118 F(n) F(n + 1) - 62940 F(n) F(n + 1) + 30525 F(n) F(n + 1) 8 3 - 22810 F(n) F(n + 1) + 236 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 867 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(4 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5385 20274 19584 5 25770 8 G(n) = ----- F(n) - ----- F(n + 1) + ----- F(n + 1) + ----- F(n + 1) 12122 6061 6061 3857 28985 8 1984 2 2 42095 7 + ----- F(n) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 15428 203 3857 280423 7 130209 3 2 42967 2 6 + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 7714 12122 15428 3207 2 3 36381 4 54468 3 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 551 6061 3857 903349 2743 4 - ------ - ---- F(n + 1) 169708 2204 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 868 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(4 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 9411 5 457085707 6 3531183 G(n) = ---- F(n + 1) - --------- F(n + 1) + ------- F(n) F(n + 1) 3781 83182 41591 6559225 2 8 4515 101691 + ------- F(n) F(n + 1) + ---- F(n) - ------ F(n + 1) 4378 7562 41591 34552453 2 12937750 10 72573 2 - -------- F(n + 1) + -------- F(n + 1) - ----- F(n) 83182 2189 7562 172318791 5 478053 4 45261 3 2 + --------- F(n) F(n + 1) - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 83182 83182 7562 63506583 3 3 53088 2 3 + -------- F(n) F(n + 1) - ----- F(n) F(n + 1) 41591 41591 69471551 2 4 49498975 3 7 - -------- F(n) F(n + 1) + -------- F(n) F(n + 1) 41591 4378 64827475 9 - -------- F(n) F(n + 1) 4378 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 869 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 690 9 2385 5 92019 4410 2 4 G(n) = ---- F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) + ---- F(n) F(n + 1) 6061 319 12122 319 15975 4 2 3780661 2 3 4236105 8 - ----- F(n) F(n + 1) - ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 638 12122 6061 8366117 4 444465 4 5 703005 4 + ------- F(n) F(n + 1) - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 12122 319 12122 1297165 3 6 3735 3 3 861580 3 2 - ------- F(n) F(n + 1) + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) 12122 319 6061 20964895 2 7 19287 2 690 2 + -------- F(n) F(n + 1) - ----- F(n) - ---- F(n + 1) 12122 12122 6061 4566 + ---- F(n) F(n + 1) 551 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 870 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(4 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 66885 2 5706157 2 1519005 3 12336 3 G(n) = ------ F(n) - ------- F(n + 1) - ------- F(n + 1) + ----- F(n) 6061 12122 12122 6061 108850 10 79875 7 590085 2 + ------ F(n + 1) + ----- F(n + 1) + ------ F(n) F(n + 1) 19 638 12122 5794433 6 927845 3 3 947551 2 4 - ------- F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 1102 551 551 547625 9 410675 3 7 31175 2 8 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 38 38 19 178425 3 4 1194861 2107493 5 + ------ F(n) F(n + 1) + ------- F(n) F(n + 1) + ------- F(n) F(n + 1) 638 12122 1102 95175 6 11925 2 5 174189 2 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 319 638 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 871 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(n + j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 372 6 1042 5 1246 3 4 G(n) = --- F(n) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 319 3801 3 3 2351 5 2 3794 6 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 687 1165 2 215 5 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 319 638 319 175 6 221 7 20 3 2963 3 - --- F(n) F(n + 1) - --- F(n) - --- F(n + 1) + ---- F(n) 638 58 319 638 20 2 151 4 2 + --- F(n + 1) + --- F(n) F(n + 1) 319 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 872 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(n + j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 6 G(n) = -1/420 F(n) (1740 F(n) - 7725 F(n) F(n + 1) - 363 F(n + 1) 7 4 3 - 4911 F(n + 1) - 1785 F(n) F(n + 1) + 4641 F(n + 1) 5 2 6 2 3 + 14028 F(n) F(n + 1) + 10500 F(n) F(n + 1) + 2695 F(n) F(n + 1) 4 3 5 2 - 18480 F(n) F(n + 1) + 934 F(n) - 1414 F(n) - 273 F(n) F(n + 1) 5 + 413 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 873 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 791 7 120 2 6 3348 3 G(n) = - --- - 265/2 F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 29 319 84575 3 1123 3 108149 4 6 + ----- F(n + 1) - ---- F(n) - ------ F(n + 1) + 625/2 F(n) F(n + 1) 638 638 638 3 4 4950 8 8291 2 - 305 F(n) F(n + 1) + ---- F(n + 1) + ---- F(n) F(n + 1) 29 638 17773 2 2 2 5 10630 3 5 - ----- F(n) F(n + 1) + 55/2 F(n) F(n + 1) + ----- F(n) F(n + 1) 638 29 44795 3 11955 7 15713 2 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 29 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 874 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 5 5 3 2 G(n) = -1/84 F(n) (-3738 F(n) F(n + 1) - 168 F(n + 1) - 994 F(n) F(n + 1) 6 3 7 + 6405 F(n) F(n + 1) + 2289 F(n + 1) - 2385 F(n + 1) 4 7 6 + 322 F(n) F(n + 1) + 369 F(n) - 3645 F(n) F(n + 1) 2 2 3 + 2163 F(n) F(n + 1) + 644 F(n) F(n + 1) + 262 F(n + 1) 2 - 1491 F(n) F(n + 1) - 33 F(n)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 875 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(n + 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 278191 2 14727 2 87459 690123 G(n) = ------- F(n + 1) - ----- F(n) + ----- F(n) - ------ F(n + 1) 12122 12122 12122 12122 577 5 629 3 3 26475 9 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + ----- F(n + 1) 11 11 19 1472617 5 501 6 43443 - ------- F(n + 1) + --- F(n + 1) + ----- F(n) F(n + 1) 1102 22 6061 66225 8 204354 2 3 233 2 4 - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) - --- F(n) F(n + 1) 19 551 22 101825 3 6 12775 2 7 141133 3 2 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 38 38 551 636043 4 + ------ F(n) F(n + 1) 1102 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 876 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(n + 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 21955 3 20 3 1041 6 393 3 G(n) = ----- F(n) + --- F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 1276 319 1276 44 5725 7 7317 6 4703 3 5 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 1276 319 1276 1399 3 105 7 9550 6 2 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 319 319 25825 7 617 8 3167 2 18930 4 3 - ----- F(n) - --- F(n) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 1276 638 1276 319 20 8 1025 5 3 375 3 4 - --- F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 319 44 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 877 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(n + 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 63275 3 6 7 3 4 G(n) = ----- F(n) F(n + 1) - 135/2 F(n + 1) - 925/6 F(n) F(n + 1) 14 6 4925 2 7 8511 3 2 + 955/6 F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 14 28 61769 5 9 1731 199 - ----- F(n + 1) + 15875/7 F(n + 1) - ---- F(n + 1) + --- F(n) 28 28 28 3 31 3 711 2 4 + 135/2 F(n + 1) - -- F(n) - --- F(n) F(n + 1) + 6631/7 F(n) F(n + 1) 28 28 8 593 2 14453 2 3 - 11225/2 F(n) F(n + 1) + --- F(n) F(n + 1) - ----- F(n) F(n + 1) 84 28 2 5 + 40/3 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 878 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 1149 9 975 7 15651 8 G(n) = ---- F(n) - --- F(n) F(n + 1) + ----- F(n) F(n + 1) 638 22 3190 3 6229 2 6 10939 2 7 + 80 F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 44 319 1297 3 5 38065 3 6 371 7 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - --- F(n) F(n + 1) 11 638 22 1295 7 2 2580 8 901 4 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 58 319 290 937 4 20 2463 2 2 + --- F(n + 1) - --- F(n + 1) - ---- - 72 F(n) F(n + 1) 44 319 116 98719 6 3 1119 8 72643 2 3 - ----- F(n) F(n + 1) + ---- F(n) + ----- F(n) F(n + 1) 1595 44 3190 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 879 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 100541 6 6589 2 2 10 G(n) = ------- F(n + 1) - ---- F(n + 1) - 9 F(n) + 7142 F(n + 1) 15 15 3 3 9 2 8 + 1664 F(n) F(n + 1) - 17848 F(n) F(n + 1) + 1664 F(n) F(n + 1) 5 2 4 + 13001/5 F(n) F(n + 1) - 1938 F(n) F(n + 1) + 429/5 F(n) F(n + 1) 3 7 + 13781 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 880 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(n + 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 63275 4 5 8539 3 2 1975 3 4 G(n) = ------ F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 28 28 12 225 9 2 5 25 3 3 + --- F(n + 1) + 55/6 F(n) F(n + 1) - -- F(n) + 295/4 F(n + 1) 28 21 691 2 401 2 1745 151 + --- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n + 1) + --- F(n) 84 14 28 21 5 73125 2 7 14439 2 3 + 380/7 F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 28 28 8 6 13241 4 - 7650/7 F(n) F(n + 1) + 1055/6 F(n) F(n + 1) + ----- F(n) F(n + 1) 14 7 - 295/4 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 881 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(n + 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3361139 6 108850 10 445 6 G(n) = -------- F(n + 1) + ------ F(n + 1) - --- F(n) F(n + 1) 551 19 22 570135 2 410675 3 7 44794073 + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) + -------- F(n) F(n + 1) 12122 38 12122 547625 9 173543 2 17881401 2 4 - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) + -------- F(n) F(n + 1) 38 12122 2204 2 5 2935 4 3 31175 2 8 + 235/2 F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 22 19 21671605 4 2 2250449 2 3963183 6 - -------- F(n) F(n + 1) + ------- F(n + 1) + ------- F(n) 2204 6061 2204 14369 3 43870685 2 145 7 78825 3 + ----- F(n) - -------- F(n) - --- F(n + 1) + ----- F(n + 1) 6061 24244 11 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 882 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(n + 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 9 5 4 5 2 G(n) = 1/672 F(n) (22848 F(n) - 219708 F(n) F(n + 1) + 14728 F(n) F(n + 1) 4 5 4 3 6 + 191059 F(n) F(n + 1) - 72858 F(n) F(n + 1) - 61524 F(n) F(n + 1) 3 4 6 8 + 448 F(n) F(n + 1) + 14728 F(n) F(n + 1) + 171527 F(n) F(n + 1) 6 5 3 7 - 7168 F(n) F(n + 1) - 6554 F(n + 1) + 6776 F(n + 1) - 7112 F(n + 1) 4 3 7 5 - 25536 F(n) F(n + 1) + 1568 F(n) - 30464 F(n) + 4051 F(n + 1) 9 + 3191 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 883 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(2 n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 1470 22125 9 7320 4 G(n) = -670 F(n + 1) + ---- + ----- F(n + 1) + ---- F(n + 1) 319 29 11 2 6 4579 444 434 3 - 15 F(n) F(n + 1) - ---- F(n + 1) + --- F(n) - --- F(n) F(n + 1) 319 319 11 238816 5 7 3 5 - ------ F(n + 1) + 1630 F(n) F(n + 1) - 1410 F(n) F(n + 1) 319 3110 3 4478 2 3 2100 2 7 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 29 29 25007 3 2 45250 3 6 2 2 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + 113 F(n) F(n + 1) 319 29 101866 4 54325 8 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 319 29 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 884 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 4715 6 2475 9 G(n) = 95/9 F(n + 1) - ---- F(n + 1) - ---- F(n) F(n + 1) 14 14 7 125 8 2 3 - 440/9 F(n) F(n + 1) - --- F(n) + 1452/7 F(n) + 136/3 F(n) F(n + 1) 18 9 25145 8 2 7 3 - 5375/7 F(n) F(n + 1) - ----- F(n) F(n + 1) + 3678/7 F(n) F(n + 1) 28 7 9001 2 8 2 6 - 436/9 F(n) F(n + 1) + ---- F(n) F(n + 1) + 164/3 F(n) F(n + 1) 28 2213 2 4 2 2 5 + ---- F(n) F(n + 1) + 16/9 F(n) F(n + 1) + 5962/7 F(n) F(n + 1) 14 4715 2 3065 10 4 + 85/9 + ---- F(n + 1) - ---- F(n) - 20 F(n + 1) 14 14 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 885 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(2 n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 46004 2 1002316 6 2 7 G(n) = ------ F(n + 1) - ------- F(n + 1) - 375 F(n) F(n + 1) 319 319 7405 3 2 187266 3 3 405175 5 - ---- F(n) F(n + 1) + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 22 319 319 8 236500 9 6927 2 3 + 7275 F(n) F(n + 1) - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 29 11 253718 2 4 16839 16900 2 8 - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 638 29 13683 4 31753 5 4063 20233 - ----- F(n) F(n + 1) + ----- F(n + 1) - ---- F(n) + ----- F(n + 1) 11 11 638 319 3 6 188325 3 7 9 - 5950 F(n) F(n + 1) + ------ F(n) F(n + 1) - 2950 F(n + 1) 29 1679 2 95300 10 - ---- F(n) + ----- F(n + 1) 638 29 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 886 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 775 4 185 3 24 3 185 1151 2 5 G(n) = --- F(n) + --- F(n + 1) + --- F(n) - --- + ---- F(n) F(n + 1) 638 638 319 638 319 503205 4 4 1707 4 3 3620625 4 8 + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) + ------- F(n) F(n + 1) 319 319 319 59810 7 604 6 4981 3 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 319 11 1349 2 2 1878750 11 + ---- F(n) F(n + 1) + 7/22 F(n) F(n + 1) + ------- F(n) F(n + 1) 638 319 65669 2 2 443750 3 9 46930 3 5 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 638 319 29 202 3 4 4985 3 4696875 2 10 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) - ------- F(n) F(n + 1) 319 319 319 617635 2 6 + ------ F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 887 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 2 G(n) = ) F(j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3329 4 2 6 G(n) = 22/3 F(n + 1) - ---- F(n + 1) + 25/4 F(n) F(n + 1) 12 5 2 6 + 50/3 F(n) F(n + 1) + 3/4 F(n) - 22/3 F(n + 1) 3 5 3 3 + 1175/2 F(n) F(n + 1) + 118 F(n) F(n + 1) + 47/3 F(n) F(n + 1) 3 3 7 - 58/3 F(n) F(n + 1) - 17/6 F(n) F(n + 1) - 4075/6 F(n) F(n + 1) 2 2 2 4 8 - 93/2 F(n) F(n + 1) + 3 F(n) F(n + 1) - 7/4 + 1675/6 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 888 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 109095 4 14859 3 23044 3 G(n) = ------ F(n + 1) - ----- F(n + 1) - ----- F(n) F(n + 1) 638 638 319 133525 7 1123 2 9149 2 2 + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 638 319 2 5 114975 3 5 1225 2 6 - 13/2 F(n) F(n + 1) - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 3066 3 2704 2 815 177 3 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- - --- F(n) 319 319 638 638 7 6 3 4 + 23 F(n + 1) - 109/2 F(n) F(n + 1) + 103/2 F(n) F(n + 1) 9975 8 - ---- F(n + 1) 58 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 889 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 12026 5 2268 4 615 9 615 G(n) = ------ F(n) + ---- F(n) - ----- F(n + 1) + ----- 6061 551 12122 12122 243773 4 5 26439 6 2 606509 6 3 - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 6061 551 12122 5734 4 3003 3 1239 7 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 6061 1102 1102 109809 3 6 33291 4 15750 7 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 12122 551 167344 7 2 48405 5 3 46725 4 4 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 1102 1102 9907 313846 5 4 27195 3 5 + ----- F(n) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 12122 6061 1102 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 890 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 11 13 3 3 11 5 G(n) = --- F(n + 1) + -- F(n + 1) - 1/2 F(n) + -- F(n + 1) 24 24 24 6 6 3 7 2 + 1/8 F(n) F(n + 1) - 226/3 F(n) F(n + 1) + 272/3 F(n) F(n + 1) 439 8 55 2 7 3 6 + --- F(n) F(n + 1) - -- F(n) F(n + 1) + 63/4 F(n) F(n + 1) 24 24 347 4 9 13 7 7 - --- F(n) F(n + 1) + 9/4 F(n) - -- F(n + 1) - 3/4 F(n) 12 24 5 4 4 3 5 2 - 109/6 F(n) F(n + 1) + 41/6 F(n) F(n + 1) - 47/4 F(n) F(n + 1) 91 2 + -- F(n) F(n + 1) 24 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 891 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 159 8 619 8 2225 4 141 4 11 G(n) = ---- F(n + 1) + --- F(n) - ---- F(n) + --- F(n + 1) + -- F(n + 1) 176 44 176 88 58 32 5 3875 6 2 41485 6 3 + --- F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 176 638 2875 7 26085 7 2 9415 8 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 88 319 638 130 3 6 16669 4 2735 5 4 + --- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 638 319 1339 2 3 1555 9 267 3 5 + ---- F(n) F(n + 1) - ---- F(n) + --- F(n) F(n + 1) 319 638 44 2069 5 3 5047 - ---- F(n) F(n + 1) - ---- 88 5104 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 892 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 287 2 8 8 G(n) = 19/5 F(n + 1) + --- F(n + 1) - 3/10 F(n + 1) + 11/2 F(n) 120 116 3 499 5 3 6 329 6 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - 4 F(n) - --- F(n + 1) - 7/2 15 15 90 154 4 4 3787 4 6 + --- F(n) F(n + 1) + 49/6 F(n) F(n + 1) - ---- F(n) F(n + 1) 45 72 2437 5 5 5 6 2 + ---- F(n) F(n + 1) + 181/2 F(n) F(n + 1) + 46/3 F(n) F(n + 1) 90 6 4 7 7 3 - 131/4 F(n) F(n + 1) + 8 F(n) F(n + 1) + 108 F(n) F(n + 1) 9767 8 2 91 10 - ---- F(n) F(n + 1) + -- F(n + 1) 72 72 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 893 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 67 2 3835 97923 6 95553 10 G(n) = -- F(n + 1) - ---- F(n + 1) + ----- F(n) - ----- F(n) 29 1914 638 638 104194 9 32653 3 3 228 9 - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) - --- F(n) 319 1276 319 7787 3 6 3 2 1291 9 + ---- F(n) F(n + 1) + 47/4 F(n) F(n + 1) + ---- F(n) F(n + 1) 1276 319 6971 8 66762 8 2 3010 8 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 1914 319 957 285001 7 3 6151 7 2 92551 6 3 + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 1276 348 1914 100269 5 5 44459 5 4 1135 3 7 - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 1276 3828 1276 1640 5 1289 6 + ---- F(n + 1) - ---- F(n + 1) 957 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 894 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 2 4 6 G(n) = 1/24 F(n) (810 F(n) F(n + 1) + 51 F(n + 1) + 10638 F(n) F(n + 1) 4 4 3 7 3 5 - 1520 F(n) F(n + 1) - 5754 F(n) F(n + 1) + 1312 F(n) F(n + 1) 3 3 3 2 8 + 3824 F(n) F(n + 1) - 232 F(n) F(n + 1) + 525 F(n) F(n + 1) 2 6 6 9 - 424 F(n) F(n + 1) - 17 F(n + 1) - 944 F(n) F(n + 1) 8 2 7 3 7 + 2117 F(n) F(n + 1) - 11250 F(n) F(n + 1) + 792 F(n) F(n + 1) - 38 4 10 8 - 64 F(n) + 150 F(n) + 24 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 895 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 5365 9 526835 8 91 2655 5 G(n) = ---- F(n + 1) + ------ F(n + 1) + -- - ---- F(n + 1) - 3/22 F(n) 22 22 22 11 30 12 547 2 3 - -- F(n + 1) - 55625/2 F(n + 1) - --- F(n) F(n + 1) 11 11 5779 2 2 770625 11 10845 3 6 + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 22 11 22 128660 3 5 219 3 2 181875 2 10 - ------ F(n) F(n + 1) + --- F(n) F(n + 1) - ------ F(n) F(n + 1) 11 11 22 265 2 7 2 6 12099 3 + --- F(n) F(n + 1) + 8610 F(n) F(n + 1) - ----- F(n) F(n + 1) 11 11 574375 3 9 13185 8 61520 7 - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 11 22 11 2341 4 569 3 42477 4 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) + ----- F(n + 1) 22 11 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 896 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 153 2 350 3 5 G(n) = -5/22 F(n + 1) + --- F(n) F(n + 1) + 5/22 - --- F(n) F(n + 1) 22 11 21 4 3 575 2 2 18 2 + -- F(n) - 3/22 F(n) + --- F(n) F(n + 1) - -- F(n) F(n + 1) 11 11 11 1550 7 6 1546 3 + ---- F(n) F(n + 1) - 7 F(n) F(n + 1) - ---- F(n) F(n + 1) 11 11 158 3 3625 2 6 2 5 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) + 13 F(n) F(n + 1) 11 11 4 4 4 3 3 4 + 325 F(n) F(n + 1) - 21 F(n) F(n + 1) + 6 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 897 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(4 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 9216 7 288 8 872 8 13592 4 5 G(n) = ----- F(n) F(n + 1) + --- F(n + 1) + --- F(n) - ----- F(n) F(n + 1) 319 319 319 319 120 9 7959 7 2 1863 8 - --- F(n) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 638 14528 5 3 17792 4 4 16044 5 4 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 319 7549 3 6 10240 3 5 1517 2 7 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 638 14878 6 3 14048 6 2 832 4 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - --- F(n + 1) 319 319 319 489 5 524 449 + --- F(n + 1) + --- - --- F(n + 1) 638 319 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 898 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(4 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 5 3 G(n) = -1/28 F(n) (1050 F(n) F(n + 1) + 231 F(n) - 2366 F(n + 1) - 63 F(n) 3 6 3 4 3 2 + 259350 F(n) F(n + 1) - 10050 F(n) F(n + 1) + 12985 F(n) F(n + 1) 2 3 3 2 - 25858 F(n) F(n + 1) + 4374 F(n + 1) + 438 F(n) F(n + 1) 8 6 4 - 313250 F(n) F(n + 1) + 10275 F(n) F(n + 1) + 53557 F(n) F(n + 1) 2 7 5 7 + 12950 F(n) F(n + 1) - 125062 F(n + 1) - 4350 F(n + 1) 2 9 - 1611 F(n) F(n + 1) + 127400 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 899 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(4 n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 4 525 4 5 135 2 6 G(n) = -443/8 F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 11 11 181 4 123 8 927 9 93 3 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) - -- F(n) F(n + 1) 44 88 88 22 279 98 5 285 5 3 105 6 2 - --- F(n) - -- F(n) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 88 11 22 11 1301 7 2 453 8 7 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) + 5/2 F(n) F(n + 1) 44 11 9 8 8 15 2 2 + 5/22 F(n + 1) - 5/22 F(n + 1) - 5/11 F(n) + -- F(n) F(n + 1) 11 50 7 + -- F(n) F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 900 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(4 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 438092 5 212 2 17465 3 6 G(n) = ------ F(n) F(n + 1) + --- F(n) - ----- F(n) F(n + 1) 319 319 319 1060655 2 8 6026 3 2 65232 3 3 + ------- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 29 319 436340 8 438665 9 1053 + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 319 937230 4 5 941030 4 6 178980 2 3 + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 319 319 319 182401 2 4 1054530 2 7 15990 3 7 - ------ F(n) F(n + 1) - ------- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 319 20352 4 16916 4 2 434717 4 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 319 319 319 20 9 2126 20 10 - --- F(n + 1) - ---- F(n) + --- F(n + 1) 319 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 901 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(4 n - j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 101 6 145 9 5 4 6 G(n) = ---- F(n + 1) - --- F(n + 1) + 7/22 F(n) + 915/8 F(n) F(n + 1) 44 176 695 4 5 9353 5 2617 4 2 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 44 220 44 947 4 409 3 7 215 3 6 - --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 176 22 44 5425 6 3 50763 5 5 745 5 4 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + --- F(n) F(n + 1) 176 220 88 13853 6 4 2565 7 2 14033 6 1423 10 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) + ---- F(n) 88 88 440 40 115 5 113 10 125 109 2 + --- F(n + 1) + --- F(n + 1) - --- F(n + 1) + --- F(n + 1) 88 88 176 88 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 902 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(4 n - j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 1358 11 2753 10 2831 7 1351 6 371 6 G(n) = ---- F(n) - ---- F(n) - ---- F(n) + ---- F(n) + --- F(n) F(n + 1) 11 66 22 33 22 5 5 3061 10 316 2 + 208/3 F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n + 1) 11 33 6533 6 5 3376 7 3 195 10 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 22 33 22 6 1303 5 405 2 5 - 117/2 F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 55 11 4867 9 274 3 647 6 553 7 - ---- F(n) F(n + 1) - --- F(n + 1) - --- F(n + 1) + --- F(n + 1) 66 11 66 22 2282 5 6 841 2 4 1603 9 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 55 330 6125 9 2 16037 6 4 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 22 110 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 903 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 29 13 3 89 2 2 G(n) = 5/22 - 5/22 F(n + 1) + -- F(n) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 22 51 3 3 2 16 4 4 - -- F(n) F(n + 1) + 4 F(n) F(n + 1) + -- F(n) - 3 F(n) F(n + 1) 11 11 5 - 2 F(n) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 904 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 G(n) = -1/12 F(n) (-2 F(n + 1) + F(n)) (15 F(n) - 30 F(n) F(n + 1) 2 2 2 3 + 21 F(n) F(n + 1) - 3 F(n) - 6 F(n) F(n + 1) + 2 F(n) F(n + 1) 4 2 + 3 F(n + 1) - 2 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 905 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3508 5 2 331 2 7019 4 3 G(n) = ---- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 638 4945 6 314 3 61 2 2 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) - -- F(n) F(n + 1) 319 319 29 828 3 7973 3 4 1116 2 5 20 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- 319 638 319 319 571 4 1887 3 20 3 - --- F(n) + ---- F(n) + --- F(n + 1) 638 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 906 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 18 5 113 2 3 G(n) = -3/2 F(n) F(n + 1) + -- F(n) F(n + 1) + --- F(n) F(n + 1) 11 22 12 2 4 90 3 2 54 4 - -- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 11 11 11 65 4 2 5 30 2 16 5 35 6 - -- F(n) F(n + 1) + 6 F(n) F(n + 1) - -- F(n) - -- F(n) + -- F(n) 11 11 11 11 2 + 5/22 F(n + 1) - 5/22 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 907 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 533 2 G(n) = 11/6 F(n) + --- F(n) F(n + 1) + 1/6 F(n) - 5/12 F(n + 1) 12 4 161 2 2 3 - 5/4 F(n) F(n + 1) - --- F(n) F(n + 1) + 5/12 F(n) F(n + 1) 12 2 5 3 2 3 - 11 F(n) F(n + 1) + 5/6 F(n) F(n + 1) - 112 F(n + 1) 6 3 4 7 - 268 F(n) F(n + 1) + 246 F(n) F(n + 1) + 112 F(n + 1) 5 + 5/12 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 908 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3115 5989 2 3 12257 4 G(n) = - ---- - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 638 638 7 27789 2 2 8 1201 - 930 F(n) F(n + 1) - ----- F(n) F(n + 1) + 745/2 F(n + 1) + ---- F(n) 319 638 543 5933 5 17489 3 2 - --- F(n + 1) + ---- F(n + 1) + ----- F(n) F(n + 1) 58 638 638 3 5 2 6 11964 3 + 720 F(n) F(n + 1) + 85 F(n) F(n + 1) + ----- F(n) F(n + 1) 319 117250 4 55478 3 - ------ F(n + 1) + ----- F(n) F(n + 1) 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 909 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 25 7 415 6 2 173 2 230 5 G(n) = --- F(n) + --- F(n) + 5/22 F(n + 1) - --- F(n) + --- F(n) F(n + 1) 22 22 11 11 369 2 123 2 3 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - 5/22 F(n + 1) 55 55 505 3 3 5 2 73 6 - --- F(n) F(n + 1) + 18 F(n) F(n + 1) + -- F(n) F(n + 1) 11 55 20 5 953 6 2 4 - -- F(n) F(n + 1) + --- F(n) F(n + 1) + 45/2 F(n) F(n + 1) 11 110 184 2 5 - --- F(n) F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 910 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 7 1063 2 2 G(n) = 59/2 F(n) F(n + 1) - 1245 F(n) F(n + 1) - ---- F(n) F(n + 1) 12 37 2 6 2 - 1/6 F(n) F(n + 1) - -- + 25 F(n) F(n + 1) + 1/12 F(n) 12 3 6083 4 3 5 + 1315/6 F(n) F(n + 1) - ---- F(n + 1) + 1060 F(n) F(n + 1) 12 8 + 510 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 911 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2247 2 3 4 34 3 G(n) = ----- F(n) F(n + 1) - 5/22 - 125/2 F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 2 5 2 2 653 2 - 225 F(n) F(n + 1) - 7/2 F(n) F(n + 1) + --- F(n) F(n + 1) 22 6 5 3 4 3 + 100 F(n) F(n + 1) + 30 F(n) F(n + 1) + 525/2 F(n) F(n + 1) 3 5 20 3 4 4 20 4 + 20 F(n) F(n + 1) + -- F(n) F(n + 1) - 45 F(n) F(n + 1) + -- F(n) 11 11 101 3 3 - --- F(n) + 5/22 F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 912 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 61 3337 6 789 3 4 2973 5 2 G(n) = - --- - ---- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 638 638 319 1613 7 1191 7 714 3 431 4 63 4 + ---- F(n + 1) + ---- F(n) - --- F(n + 1) - --- F(n) - --- F(n + 1) 638 638 319 638 638 799 2 2 736 3 149 4 3 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 638 353 6 - --- F(n) F(n + 1) 58 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 913 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4955 2 7 12777 2 2 14226785 2 3 G(n) = ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - -------- F(n) F(n + 1) 22 6061 24244 17628 3 8944329 3 2 28935 3 6 + ----- F(n) F(n + 1) + ------- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 12122 11 6825279 4 45795 7660517 2733 + ------- F(n) F(n + 1) + ----- F(n) - ------- F(n + 1) - ---- 24244 12122 24244 6061 3963 4 441 4 24533183 5 29215 9 - ----- F(n) + ---- F(n + 1) - -------- F(n + 1) + ----- F(n + 1) 12122 1102 24244 22 8 - 6585/2 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 914 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 2 2 G(n) = -1/12 F(n) (-31 F(n) F(n + 1) - 14 F(n) F(n + 1) + 38 F(n) F(n + 1) 2 4 4 2 - 282 F(n) F(n + 1) + 7 F(n + 1) + 147 F(n) F(n + 1) - 513 F(n + 1) 2 4 3 3 6 - 24 F(n) + 12 F(n) + 1302 F(n) F(n + 1) + 504 F(n + 1) 5 - 1146 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 915 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 97037 4 1618 5 8 2 6 G(n) = ------ F(n + 1) + ---- F(n + 1) + 615/2 F(n + 1) + 45 F(n) F(n + 1) 319 319 15605 3 88423 3 4133 2 3 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 638 638 39889 2 2 7 3 5 963 - ----- F(n) F(n + 1) - 760 F(n) F(n + 1) + 615 F(n) F(n + 1) - --- 638 319 5134 3 2 3562 4 325 311 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) - --- F(n + 1) 319 319 319 58 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 916 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 3 4 G(n) = -1/12 F(n) (140 F(n) F(n + 1) + 5970 F(n) F(n + 1) 6 2 7 - 6180 F(n) F(n + 1) - 270 F(n) F(n + 1) + 2610 F(n + 1) 2 3 3 + 978 F(n) F(n + 1) - 2622 F(n + 1) - 15 F(n) + 39 F(n) 3 2 2 5 - 290 F(n) F(n + 1) - 540 F(n) F(n + 1) + 90 F(n + 1) 4 5 + 170 F(n) F(n + 1) - 80 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 917 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 9 3251 2 7 G(n) = 2605/2 F(n + 1) + ---- F(n) F(n + 1) + 280 F(n) F(n + 1) 319 6 5 1203 2 3 3 + 65/2 F(n + 1) - 75 F(n) F(n + 1) - ---- F(n) + 80 F(n) F(n + 1) 638 803897 5 352613 4 8 - ------ F(n + 1) + ------ F(n) F(n + 1) - 6495/2 F(n) F(n + 1) 638 638 2 4 103871 2 3 3 6 - 15 F(n) F(n + 1) - ------ F(n) F(n + 1) + 5065/2 F(n) F(n + 1) 319 130623 3 2 3117 26913 10460 2 + ------ F(n) F(n + 1) + ---- F(n) - ----- F(n + 1) - ----- F(n + 1) 638 638 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 918 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 3 5 4 G(n) = -725/6 F(n + 1) - 9/4 F(n) + 2045/2 F(n) F(n + 1) 4 5 3 6 3 4 - 135 F(n) F(n + 1) - 1005 F(n) F(n + 1) - 1625/6 F(n) F(n + 1) 2 3 163 2 6 - 1897/4 F(n) F(n + 1) + --- F(n) F(n + 1) + 575/2 F(n) F(n + 1) 12 563 2 5 - --- F(n) F(n + 1) + 199/4 F(n + 1) + 21/4 F(n) - 199/4 F(n + 1) 12 3 2 5 2 7 + 725/6 F(n + 1) + 50/3 F(n) F(n + 1) + 865/2 F(n) F(n + 1) 3 2 4 + 1039/4 F(n) F(n + 1) - 207/2 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 919 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 814807 7 2 21987 2 7 727 2 6 G(n) = ------ F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 12122 6061 6061 19866 3 6 8475 2 2 6869 2 3 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 551 6061 1102 13612 3 284357 3 2 159718 8 - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 6061 12122 6061 6592 3 5350 4 4 7663 1070 8 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) + ---- F(n + 1) 6061 6061 6061 6061 5479 8 11102 9 1070 9 8818 3 5 + ---- F(n) + ----- F(n) - ---- F(n + 1) - ---- F(n) F(n + 1) 6061 6061 6061 6061 719063 4 5 11386 7 - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 12122 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 920 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 6 G(n) = -1/924 F(n) (-78 F(n) F(n + 1) + 234 F(n) F(n + 1) 6 3 7 3 9 - 219936 F(n) F(n + 1) + 39 F(n) - 52 F(n + 1) + 5505 F(n) 8 5 2 3 4 + 32206 F(n) F(n + 1) + 351 F(n) F(n + 1) - 351 F(n) F(n + 1) 3 6 4 8 + 22788 F(n) F(n + 1) - 67016 F(n) F(n + 1) - 2438 F(n) F(n + 1) 2 3 2 5 - 260 F(n) F(n + 1) + 7499 F(n) F(n + 1) - 122 F(n + 1) 7 2 4 6 + 218810 F(n) F(n + 1) + 2743 F(n) F(n + 1) + 78 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 921 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 1213 4 4787 3 2 1433 G(n) = 615/2 F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- 319 638 638 2 6 13691 3 2857 2 3 + 45 F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 638 38613 2 2 7 89061 3 - ----- F(n) F(n + 1) - 760 F(n) F(n + 1) + ----- F(n) F(n + 1) 638 638 3 5 157 1449 97356 4 + 615 F(n) F(n + 1) + --- F(n) - ---- F(n + 1) - ----- F(n + 1) 638 638 319 1409 5 + ---- F(n + 1) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 922 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1070 9 3877 1070 10 68855 2 G(n) = ---- F(n + 1) - ----- F(n) - ---- F(n + 1) - ----- F(n) 6061 12122 6061 12122 358525 13848 4 4729105 3 7 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ------- F(n) F(n + 1) 6061 6061 12122 2140 3 6 1070 4 5 3875015 4 2 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ------- F(n) F(n + 1) 6061 6061 12122 33758245 9 2140 8 33846245 4 6 - -------- F(n) F(n + 1) - ---- F(n) F(n + 1) - -------- F(n) F(n + 1) 12122 6061 6061 10451 2 3 1070 2 7 40031 3 2 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 6061 12122 14391867 2 4 83409765 2 8 - -------- F(n) F(n + 1) + -------- F(n) F(n + 1) 12122 12122 333299 3 3 33057537 5 6043 4 + ------ F(n) F(n + 1) + -------- F(n) F(n + 1) - ----- F(n) F(n + 1) 551 12122 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 923 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 6995 3 2 56495 4 5 18377 6 3 G(n) = ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 2552 1276 20209 7 2 99219 8 595 3 3 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - --- F(n) F(n + 1) 319 2552 319 90 4 2 805 6 235 6 151 41 2 - -- F(n) F(n + 1) + --- F(n) + --- F(n + 1) + --- F(n + 1) - -- F(n) 11 319 638 88 11 25 2 345 5 2164 5 3585 5 - -- F(n + 1) - --- F(n + 1) + ---- F(n) + ---- F(n) F(n + 1) 58 116 319 319 504 9 3371 9 - --- F(n) + ---- F(n + 1) 319 2552 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 924 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 158002 6 925 7 320757 2 42 2 G(n) = ------ F(n) + --- F(n) - ------ F(n) - --- F(n) F(n + 1) 319 638 638 319 20 3 1383199 5 475 6 - --- F(n + 1) - ------- F(n) F(n + 1) + --- F(n) F(n + 1) 319 1276 638 853125 9 2653 2 5032025 3 7 + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) - ------- F(n) F(n + 1) 1276 638 1276 495659 4 2 5275 4 3 4 6 - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) + 2875/2 F(n) F(n + 1) 319 319 207354 5 475 5 2 4362285 5 5 + ------ F(n) F(n + 1) + --- F(n) F(n + 1) + ------- F(n) F(n + 1) 319 22 1276 2275 6 132266 20 10 - ---- F(n) F(n + 1) + ------ F(n) F(n + 1) + --- F(n + 1) 638 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 925 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 22752459 3 1041957 4 701484534 2 G(n) = -------- F(n) - ------- F(n) + --------- F(n) F(n + 1) 2412278 2412278 1206139 23778 3 2559631183 6 + ------- F(n) F(n + 1) + ---------- F(n) F(n + 1) 1206139 219298 503145 7 509663555 10 + ------- F(n) F(n + 1) - --------- F(n) F(n + 1) 1206139 41591 245956875 2 1545102 2 2 - --------- F(n) F(n + 1) - ------- F(n) F(n + 1) 2412278 1206139 519633896 2 5 503145 2 6 - --------- F(n) F(n + 1) + ------- F(n) F(n + 1) 109649 2412278 73602711455 2 9 7505026301 3 4 + ----------- F(n) F(n + 1) + ---------- F(n) F(n + 1) 2412278 2412278 503145 3 5 3009740005 3 8 - ------- F(n) F(n + 1) - ---------- F(n) F(n + 1) 1206139 1206139 5692488291 4 3 503145 4 4 - ---------- F(n) F(n + 1) - ------- F(n) F(n + 1) 2412278 2412278 57737068305 4 7 503145 8 503145 11 - ----------- F(n) F(n + 1) - ------- F(n + 1) + ------- F(n + 1) 2412278 2412278 2412278 1053846 3 + ------- F(n) F(n + 1) 1206139 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 926 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7583731 3 3 237125 3 7 738539 G(n) = ------- F(n) F(n + 1) + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 6061 22 12122 25174165 5 28525 2 8 1398 3 2 + -------- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 12122 22 551 2049 4 3255 2 1960374 2 7017 - ---- F(n) F(n + 1) - ---- F(n) - ------- F(n + 1) - ----- F(n + 1) 6061 551 6061 12122 51 291 5 63768987 6 306975 9 - --- F(n) + --- F(n + 1) - -------- F(n + 1) - ------ F(n) F(n + 1) 551 551 12122 22 33501 2 3 9115258 2 4 61425 10 - ----- F(n) F(n + 1) - ------- F(n) F(n + 1) + ----- F(n + 1) 12122 6061 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 927 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(4 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 3 G(n) = -1/924 F(n) (565 - 3048461 F(n) F(n + 1) - 2375 F(n) F(n + 1) 2 2 6 2 8 + 1925 F(n) F(n + 1) - 11627363 F(n + 1) + 2388925 F(n) F(n + 1) 3 3 2 2 + 2270651 F(n) F(n + 1) - 554326 F(n + 1) - 8881 F(n) 3 7 9 3 + 23868075 F(n) F(n + 1) - 30303350 F(n) F(n + 1) + 900 F(n) F(n + 1) 5 4 10 + 4736362 F(n) F(n + 1) - 450 F(n + 1) + 12181400 F(n + 1) + 96403 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 928 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(4 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 125341759 7 70775 2 9 83537561 2 5 G(n) = ---------- F(n + 1) + ----- F(n) F(n + 1) - -------- F(n) F(n + 1) 6061 11 12122 46832248 3 4 240 3 3 45 2 4 + -------- F(n) F(n + 1) + --- F(n) F(n + 1) - -- F(n) F(n + 1) 6061 11 11 958325 3 8 79405597 6 225 5 + ------ F(n) F(n + 1) + -------- F(n) F(n + 1) - --- F(n) F(n + 1) 22 12122 11 3526731 2 19263 43227 2 + ------- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 6061 418 636550 10 506525 11 117405 3 8307 2 - ------ F(n) F(n + 1) + ------ F(n + 1) + ------ F(n) - ----- F(n) 11 22 12122 12122 195 6 54030 2 14205571 3 + --- F(n + 1) - ----- F(n + 1) - -------- F(n + 1) 22 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 929 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 5 2 4 G(n) = 5/22 F(n + 1) + 7/11 F(n) + 4 F(n) F(n + 1) - 6 F(n) F(n + 1) 4 2 6 47 2 + 22 F(n) F(n + 1) - 8 F(n) F(n + 1) - -- F(n) F(n + 1) 22 2 5 3 3 3 4 + 17 F(n) F(n + 1) - 17 F(n) F(n + 1) + 14 F(n) F(n + 1) 4 3 64 3 3 - 29 F(n) F(n + 1) - -- F(n) F(n + 1) + 4/11 F(n) - 5/22 F(n + 1) 11 219 2 + --- F(n) F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 930 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(n + j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 35 1007 2 2 3 5 G(n) = - -- - ---- F(n) F(n + 1) - 5 F(n) F(n + 1) + 1040 F(n) F(n + 1) 12 12 2 4 7 5 + 11/2 F(n) F(n + 1) - 2425/2 F(n) F(n + 1) + 209/6 F(n) F(n + 1) 6 3 3 2 6 11 2 - 91/6 F(n + 1) - 223/6 F(n) F(n + 1) + 35/2 F(n) F(n + 1) + -- F(n) 12 2 3 3 + 91/6 F(n + 1) + 637/3 F(n) F(n + 1) + 55/2 F(n) F(n + 1) 5935 4 8 - ---- F(n + 1) + 995/2 F(n + 1) 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 931 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(n + j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 6 102276 2 3 144 2 4 G(n) = 5065/2 F(n) F(n + 1) - ------ F(n) F(n + 1) - --- F(n) F(n + 1) 319 11 8 1772 2 7 - 6495/2 F(n) F(n + 1) + ---- F(n) F(n + 1) + 280 F(n) F(n + 1) 319 60367 3 2 359631 4 574 3 3 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + --- F(n) F(n + 1) 319 638 11 989 5 223 6 403703 5 9 - --- F(n) F(n + 1) + --- F(n + 1) - ------ F(n + 1) + 2605/2 F(n + 1) 22 11 319 23549 2421 507 2 6487 2 - ----- F(n + 1) + ---- F(n) - --- F(n) - ---- F(n + 1) 638 638 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 932 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(n + j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 4 2 5 2 6 G(n) = -145 F(n) F(n + 1) - 105 F(n) F(n + 1) + 125 F(n) F(n + 1) 75 3 3 4 625 3 + -- F(n) F(n + 1) - 40 F(n) F(n + 1) + --- F(n) F(n + 1) 22 11 7 301 2 2 2 - 55 F(n) F(n + 1) + --- F(n) F(n + 1) - 35/2 F(n) F(n + 1) - 5/22 22 3 5 546 2 23 3 3 + 35 F(n) F(n + 1) - --- F(n) F(n + 1) - -- F(n) + 5/22 F(n + 1) 11 11 4 4 3 6 + 7/22 F(n) + 265/2 F(n) F(n + 1) + 95/2 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 933 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 4 6 G(n) = -1/24 F(n) (-14 + 46 F(n) + 15 F(n + 1) - 104 F(n) - 5 F(n + 1) 3 5 4 2 2 2 - 7140 F(n) F(n + 1) + 275 F(n) F(n + 1) - 3682 F(n) F(n + 1) 2 4 2 6 5 3 + 85 F(n) F(n + 1) + 3360 F(n) F(n + 1) + 10380 F(n) F(n + 1) 5 3 3 4 4 - 250 F(n) F(n + 1) - 120 F(n) F(n + 1) - 3780 F(n) F(n + 1) 3 + 934 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 934 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 56 2 2 2 3 G(n) = --- F(n + 1) + 3/11 F(n) + 191/2 F(n) F(n + 1) - 227/2 F(n) F(n + 1) 11 3 5 4 452 3 2 - 2675/2 F(n) F(n + 1) + 1245/2 F(n + 1) + --- F(n) F(n + 1) 11 8 30 6207 4 2 7 - 1630 F(n) F(n + 1) + -- + ---- F(n) F(n + 1) + 15 F(n) F(n + 1) 11 22 3 6 2 6 14633 5 + 1410 F(n) F(n + 1) + 25/2 F(n) F(n + 1) - ----- F(n + 1) 22 9 3 8 + 670 F(n + 1) - 57/2 F(n) F(n + 1) - 625 F(n + 1) 3 7 - 519/2 F(n) F(n + 1) + 3025/2 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 935 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 184727 6 3 446209 7 2 194998 8 G(n) = ------- F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 7975 7975 7975 7351 2798 5 2 4 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - 23/2 F(n) F(n + 1) 1595 319 59109 5 4 82317 4 3167 6 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n + 1) 15950 7975 1595 565 6 30672 5 30466 9 6307 9 - --- F(n) + ----- F(n + 1) + ----- F(n) + ----- F(n + 1) 638 7975 7975 15950 31513 16993 5409 2 14401 5 - ----- F(n + 1) + ----- F(n) + ---- F(n + 1) + ----- F(n) F(n + 1) 7975 15950 3190 1595 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 936 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(n + 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 7 2 2 G(n) = 1/12 F(n) (11 F(n + 1) + 5168 F(n) F(n + 1) + 1842 F(n) F(n + 1) 2 4 2 6 3 - 80 F(n) F(n + 1) - 12071 F(n) F(n + 1) - 469 F(n) F(n + 1) 3 3 4 4 5 + 160 F(n) F(n + 1) + 12281 F(n) F(n + 1) + 70 F(n) F(n + 1) 2 3 5 4 2 - 9 F(n + 1) - 1598 F(n) F(n + 1) - 150 F(n) F(n + 1) 3 2 4 - 5212 F(n) F(n + 1) - 9 F(n) + 57 F(n) + 9 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 937 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(n + 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 185 3 3 4 853117 5 G(n) = ---- F(n + 1) + 30 F(n) F(n + 1) + ------ F(n) F(n + 1) 638 319 27361 2 185 10 1215 3 5043 2 + ----- F(n) F(n + 1) + --- F(n + 1) + ---- F(n) - ---- F(n) 638 638 638 638 6 877435 9 749473 2 4 - 85/2 F(n) F(n + 1) - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 319 638 3445015 4 6 2181070 2 8 - ------- F(n) F(n + 1) + ------- F(n) F(n + 1) 638 319 207180 3 3 47783 7851 2 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 638 2 5 4 3 120763 4 2 + 185/2 F(n) F(n + 1) - 225/2 F(n) F(n + 1) - ------ F(n) F(n + 1) 319 167290 3 7 - ------ F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 938 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 105617 6 4 5533 2 G(n) = -950/3 F(n + 1) - ------ F(n + 1) + 315 F(n + 1) - ---- F(n + 1) 12 12 3 2 6 + 248/3 F(n) F(n + 1) - 31/2 F(n) F(n + 1) + 25/6 F(n) F(n + 1) 2 2 8 10 - 23/3 F(n) + 5/3 + 1925 F(n) F(n + 1) + 18525/2 F(n + 1) 9 2 2 2 4 - 23075 F(n) F(n + 1) + 49 F(n) F(n + 1) - 4757/2 F(n) F(n + 1) 3 7 3 5 + 36125/2 F(n) F(n + 1) - 131 F(n) F(n + 1) + 10639/3 F(n) F(n + 1) 7 3 3 3 5 + 2300/3 F(n) F(n + 1) + 5539/3 F(n) F(n + 1) - 675 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 939 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 18418351 3 4 30157755 3 8 G(n) = -------- F(n) F(n + 1) - -------- F(n) F(n + 1) 6061 12122 2140 3 7 1717 3 3 357091465 2 9 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + --------- F(n) F(n + 1) 6061 6061 12122 1070 2 8 950408 2 5 14983 2 4 - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 209 1102 43721 2 71641635 10 2140 9 - ----- F(n) F(n + 1) - -------- F(n) F(n + 1) - ---- F(n) F(n + 1) 418 6061 6061 136233391 6 35751 5 7067237 2 + --------- F(n) F(n + 1) - ----- F(n) F(n + 1) + ------- F(n) F(n + 1) 12122 6061 12122 2753 279097415 4 7 1070 4 6 + ---- F(n) F(n + 1) - --------- F(n) F(n + 1) + ---- F(n) F(n + 1) 551 12122 6061 28413897 4 3 62284 3 7735 2 1070 11 - -------- F(n) F(n + 1) + ----- F(n) - ---- F(n) - ---- F(n + 1) 12122 6061 6061 6061 72213 4 2 1070 10 - ----- F(n) F(n + 1) + ---- F(n + 1) 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 940 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(n + 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 10 20 3 11717 2 G(n) = --- F(n + 1) - --- F(n + 1) + ----- F(n) F(n + 1) 319 319 638 30055 4 3 386271 2 4 877290 9 - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 638 319 319 11065 6 857841 5 19066 - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 638 319 319 4362285 2 8 19905 2 5 4665 6 + ------- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 638 319 1722580 4 6 7467 4 2 167435 3 7 - ------- F(n) F(n + 1) - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 319 22 319 206716 3 3 290 3 4 2768 6 854 3 + ------ F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) + --- F(n) 319 11 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 941 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(n + 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 24788 10 1275 5 3 20 20 11 G(n) = ------ F(n) F(n + 1) + ---- F(n) F(n + 1) - --- + --- F(n + 1) 319 22 319 319 75 8 7957 11 17 3 206521 6 5 - -- F(n) + ---- F(n) + -- F(n) F(n + 1) - ------ F(n) F(n + 1) 22 638 22 638 1575 6 2 6033 3 8 454009 5 2 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 22 319 3190 687 2 50 2 6 315 2 5 - --- F(n) F(n + 1) - -- F(n) F(n + 1) + --- F(n) F(n + 1) 638 11 58 155467 4 7 25 4 4 7003 4 3 + ------ F(n) F(n + 1) + -- F(n) F(n + 1) + ---- F(n) F(n + 1) 3190 22 290 41796 5 6 25 7 186 3 + ----- F(n) F(n + 1) + -- F(n) F(n + 1) + --- F(n) F(n + 1) 319 22 11 1095909 9 2 + ------- F(n) F(n + 1) 3190 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 942 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 2 6 3 115 G(n) = -670 F(n + 1) - 15 F(n) F(n + 1) - 42 F(n) F(n + 1) + --- 22 3230 4 607 3 2 3 6 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) + 1410 F(n) F(n + 1) 11 22 2 3 2 7 3 5 - 111 F(n) F(n + 1) + 15 F(n) F(n + 1) - 1410 F(n) F(n + 1) 3 7 8 - 282 F(n) F(n + 1) + 1630 F(n) F(n + 1) - 1630 F(n) F(n + 1) 2 2 14743 5 27 + 115 F(n) F(n + 1) - ----- F(n + 1) - -- F(n) - 1/11 F(n + 1) 22 22 4 9 + 665 F(n + 1) + 670 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 943 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(2 n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 10 4 4 5557 4 2 G(n) = 18525/2 F(n + 1) + 2 F(n) - 145/2 F(n + 1) - ---- F(n) F(n + 1) 12 4 4 3 7 3 3 + 815/2 F(n) F(n + 1) + 36125/2 F(n) F(n + 1) + 2771/3 F(n) F(n + 1) 5 9 + 503/6 F(n) F(n + 1) + 4467 F(n) F(n + 1) - 23075 F(n) F(n + 1) 2 2 22967 2 4 2 6 + 107/2 F(n) F(n + 1) - ----- F(n) F(n + 1) - 415 F(n) F(n + 1) 12 2 8 3 6 2 + 1925 F(n) F(n + 1) - 43/3 F(n) F(n + 1) - 18525/2 F(n + 1) - 8 F(n) 8 3 3 5 + 145/2 F(n + 1) - 437/3 F(n) F(n + 1) + 110 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 944 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(2 n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 161 73155 2 7 2 4 G(n) = ---- F(n) - ----- F(n) F(n + 1) - 1161/2 F(n) F(n + 1) 22 22 12469 2 3 15120 9 15120 8 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 22 11 11 30303 5 15075 4 64905 4 6 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 22 11 22 64905 4 5 526 4 2 1501 4 + ----- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 22 11 22 545 3 7 545 3 6 2265 3 3 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 11 11 4709 3 2 73155 2 8 29 2 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) + -- F(n) 22 22 22 9 10 + 1/2 F(n) F(n + 1) + 5/22 F(n + 1) - 5/22 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 945 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 845958 3 8 7 G(n) = ------- F(n + 1) + 1085/2 F(n + 1) - 1320 F(n) F(n + 1) 319 3506901 6 3 437685 2 + ------- F(n) F(n + 1) + 228 F(n) F(n + 1) + ------ F(n) F(n + 1) 638 638 3 11 4080 3 2 6 + 32 F(n) F(n + 1) + 22025 F(n + 1) + ---- F(n) + 10 F(n) F(n + 1) 319 10 3 8 81665 2 - 111075/2 F(n) F(n + 1) + 41200 F(n) F(n + 1) - ----- F(n) F(n + 1) 638 2 2 4423253 2 5 3 5 - 92 F(n) F(n + 1) - ------- F(n) F(n + 1) + 1140 F(n) F(n + 1) 638 2701124 3 4 2 9 1209 4 + ------- F(n) F(n + 1) + 13475/2 F(n) F(n + 1) - ---- - 539 F(n + 1) 319 319 12359849 7 - -------- F(n + 1) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 946 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(2 n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 11 47 3 G(n) = 76975/2 F(n + 1) + 104/3 F(n + 1) - -- F(n) - 13811/4 F(n + 1) 12 5 4 5 3 8 - 104/3 F(n + 1) + 1475 F(n) F(n + 1) + 73400 F(n) F(n + 1) 3 6 140119 3 4 3 2 - 25 F(n) F(n + 1) + ------ F(n) F(n + 1) - 176 F(n) F(n + 1) 12 2 7 2 3 - 3325/2 F(n) F(n + 1) + 1897/6 F(n) F(n + 1) 10 8 141623 6 - 193025/2 F(n) F(n + 1) + 1375/2 F(n) F(n + 1) + ------ F(n) F(n + 1) 12 4 2 2 - 3691/6 F(n) F(n + 1) + 4987/6 F(n) F(n + 1) - 285/2 F(n) F(n + 1) 7 2 9 2 5 - 140139/4 F(n + 1) + 20075/2 F(n) F(n + 1) - 66607/6 F(n) F(n + 1) 155 3 + --- F(n) 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 947 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 11 26817 3 4 10 23 3 G(n) = -5/22 F(n + 1) + ----- F(n) F(n + 1) + 5/22 F(n + 1) - -- F(n) 22 22 221 2 276655 4 7 13558 3 3 + --- F(n) - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 22 22 11 332205 2 9 332205 2 8 28987 2 5 + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 22 22 11 56777 2 4 59 2 67645 10 + ----- F(n) F(n + 1) - -- F(n) F(n + 1) - ----- F(n) F(n + 1) 22 22 11 67645 9 133511 6 66489 5 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 11 22 11 930 2 1195 276655 4 6 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) + ------ F(n) F(n + 1) 11 11 22 12027 4 3 13381 4 2 6055 3 8 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 22 22 11 6055 3 7 + ---- F(n) F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 948 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 11 G(n) = 1/6 F(n) (-63 F(n) - 9889 F(n + 1) - 169 F(n + 1) - 222500 F(n + 1) 2 4 - 3978 F(n) F(n + 1) + 4217 F(n) F(n + 1) + 15 F(n) 2 7 2 9 3 2 + 800 F(n) F(n + 1) - 52500 F(n) F(n + 1) + 952 F(n) F(n + 1) 3 4 3 6 3 8 - 25406 F(n) F(n + 1) + 20600 F(n) F(n + 1) - 428750 F(n) F(n + 1) 8 7 9 - 24650 F(n) F(n + 1) + 222512 F(n + 1) + 10050 F(n + 1) 10 2 6 + 556250 F(n) F(n + 1) + 672 F(n) F(n + 1) - 107314 F(n) F(n + 1) 4 3 2 3 2 5 + 17092 F(n) F(n + 1) - 1989 F(n) F(n + 1) + 44048 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 949 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 109 6173 4 8 G(n) = 6/11 F(n) - --- F(n + 1) + ---- F(n) F(n + 1) - 1630 F(n) F(n + 1) 22 22 1068 2 2 2419 2 3 2 7 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + 15 F(n) F(n + 1) 11 22 426 3 2 3 6 360 3 + --- F(n) F(n + 1) + 1410 F(n) F(n + 1) - --- F(n) F(n + 1) 11 11 9 12825 8 150 2 6 38 + 670 F(n + 1) - ----- F(n + 1) - --- F(n) F(n + 1) + -- 22 11 11 6372 4 7313 5 2712 3 + ---- F(n + 1) - ---- F(n + 1) - ---- F(n) F(n + 1) 11 11 11 15600 7 13500 3 5 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 11 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 950 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 3 5 G(n) = 5672/3 F(n) F(n + 1) - 1175/2 F(n) F(n + 1) 3 7 7 + 36125/2 F(n) F(n + 1) + 272/3 F(n) F(n + 1) + 4075/6 F(n) F(n + 1) 9 2 2 2 4 - 23075 F(n) F(n + 1) + 93/2 F(n) F(n + 1) - 4761/2 F(n) F(n + 1) 2 8 5 3 + 1925 F(n) F(n + 1) + 20989/6 F(n) F(n + 1) - 47/3 F(n) F(n + 1) 2 6 2 2 3329 4 - 25/4 F(n) F(n + 1) - 2887/6 F(n + 1) - 39/4 F(n) + ---- F(n + 1) 12 6 10 8 - 26344/3 F(n + 1) + 7/4 + 18525/2 F(n + 1) - 1675/6 F(n + 1) 3 - 118 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 951 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 105 8 3336 7 635 6 G(n) = ---- F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 638 1595 319 1656 3 2111 2 70613 6 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 1595 638 638 1804173 5 6 18570 5 3 136270 5 2 - ------- F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 638 319 319 799309 4 7 874804 7 4 96342 6 5 + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 638 319 319 12597 4 4 748313 4 3 13917 3 8 - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 319 638 638 13062 2 2 54843 6 2 1173 4 5001 7 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) + ---- F(n) 1595 1595 319 319 726 7 105 11 + --- F(n) F(n + 1) + --- F(n + 1) 29 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 952 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 10 5 1635 4 6 1380 5 5 G(n) = --- F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 22 11 1445 6 3 1567 6 4 7 2 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - 98 F(n) F(n + 1) 11 11 2115 7 3 837 8 1437 8 2 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 22 22 44 137 9 113 2 8 37 6 19 10 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n + 1) - -- F(n) 22 44 44 22 157 9 15 27 2 5 4 - --- F(n) + -- F(n + 1) - -- F(n + 1) - 115 F(n) F(n + 1) 22 22 44 41 2 7 402 3 6 345 3 7 + -- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 11 11 11 1699 4 5 + ---- F(n) F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 953 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 2 7 3 G(n) = 1/600 F(n) (-65984 F(n) F(n + 1) - 741948 F(n) F(n + 1) 7 6 4 3 5 + 38475 F(n) F(n + 1) + 570692 F(n) F(n + 1) + 60900 F(n) F(n + 1) 3 2 6 2 4 - 24825 F(n) F(n + 1) - 96700 F(n) F(n + 1) + 218492 F(n) F(n + 1) 2 2 9 7 + 22150 F(n) F(n + 1) - 7316 F(n) F(n + 1) + 33125 F(n) F(n + 1) 5 3 - 69072 F(n) F(n + 1) + 75538 F(n) F(n + 1) - 33125 F(n) F(n + 1) 9 8 2 10 + 11548 F(n) F(n + 1) - 1700 F(n) - 35734 F(n) + 45184 F(n) 2 + 850 F(n + 1) - 550) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 954 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 10 7 2 11 G(n) = -12575 F(n + 1) - 11942 F(n + 1) + 611 F(n + 1) + 12575 F(n + 1) 31 3 233 2 13921 3 174 2 + -- F(n) + --- F(n) - ----- F(n + 1) - --- F(n) F(n + 1) 22 22 22 11 10 9 6 - 31300 F(n) F(n + 1) + 31300 F(n) F(n + 1) + 4763 F(n) F(n + 1) 53006 5 2703 2 1219 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 22 11 3 8 3 7 3 4 + 24600 F(n) F(n + 1) - 24600 F(n) F(n + 1) + 2489 F(n) F(n + 1) 26958 3 3 2 8 70031 2 4 - ----- F(n) F(n + 1) - 2525 F(n) F(n + 1) + ----- F(n) F(n + 1) 11 22 2 9 2 5 263203 6 + 2525 F(n) F(n + 1) - 3174 F(n) F(n + 1) + ------ F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 955 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 2 G(n) = 1/6 F(n) (45 F(n) F(n + 1) + 4 F(n + 1) - 6 F(n) F(n + 1) 4 2 2 3 + 9 F(n) F(n + 1) + 8 F(n) F(n + 1) + 6 F(n) F(n + 1) 3 2 3 - 54 F(n) F(n + 1) - 9 F(n) - 3 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 956 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 75 7 2 2 43 3 G(n) = - 5/22 + -- F(n) - 7/2 F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 169 4 3 159 6 3 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + 9/11 F(n) F(n + 1) 22 11 25 6 3 131 2 - -- F(n) F(n + 1) + 5/22 F(n + 1) + --- F(n) F(n + 1) 22 22 79 3 4 56 5 2 13 4 + -- F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) 22 11 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 957 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 567 3 84 4 32421 4101 G(n) = --- F(n) F(n + 1) + --- F(n + 1) - ----- F(n + 1) + ---- F(n) 319 319 638 638 15900 9 444837 5 8 273 + ----- F(n + 1) - ------ F(n + 1) - 3600 F(n) F(n + 1) - --- 11 319 638 3275 2 7 75198 3 2 31025 3 6 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 11 319 11 21 3 191666 4 1029 2 2 - -- F(n) F(n + 1) + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 29 319 638 115036 2 3 - ------ F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 958 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 6 2 G(n) = 1/30 F(n) (-10 F(n) + 70 F(n) + 277 F(n) F(n + 1) - 89 F(n + 1) 4 3 2 2 + 15 F(n + 1) - 30 F(n) F(n + 1) + 10 F(n) F(n + 1) 2 4 3 6 + 660 F(n) F(n + 1) + 5 F(n) F(n + 1) + 94 F(n + 1) 5 5 - 470 F(n) F(n + 1) - 532 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 959 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4033 4 18 91 5 62 G(n) = ----- F(n + 1) - 17/2 F(n + 1) + -- F(n) + -- F(n + 1) - -- 11 11 11 11 8 967 2 2 211 2 3 + 745/2 F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 11 22 288 3 2 373 4 2 6 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + 85 F(n) F(n + 1) 11 22 434 3 7 3 5 + --- F(n) F(n + 1) - 930 F(n) F(n + 1) + 720 F(n) F(n + 1) 11 1894 3 + ---- F(n) F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 960 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 3 7 G(n) = 510 F(n + 1) + 653/3 F(n) F(n + 1) - 1245 F(n) F(n + 1) 3 5 3 + 1060 F(n) F(n + 1) + 63/2 F(n) F(n + 1) + 1/3 F(n) F(n + 1) 2 2 2 6 2 - 268/3 F(n) F(n + 1) + 25 F(n) F(n + 1) - 1/6 F(n) - 23/6 4 - 3037/6 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 961 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 67 2 5 2 4 G(n) = --- F(n) - 15 F(n) F(n + 1) + 30 F(n) F(n + 1) 22 4 2 8 17093 4 - 105/2 F(n) F(n + 1) - 1575/2 F(n) F(n + 1) + ----- F(n) F(n + 1) 22 4 5 1533 4 3 6 - 3075/2 F(n) F(n + 1) - ---- F(n) F(n + 1) - 325/2 F(n) F(n + 1) 22 3 3 1863 3 2 2 7 + 25 F(n) F(n + 1) + ---- F(n) F(n + 1) + 3925/2 F(n) F(n + 1) 11 7959 2 3 183 199 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) + 5/22 F(n + 1) 22 11 22 2 - 5/22 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 962 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 7 671 9 9 G(n) = -1/30 F(n + 1) - 1/6 F(n) - --- F(n) - 5/16 F(n + 1) 16 5 2 2 3 4 - 7/30 F(n) F(n + 1) + 7/30 F(n) F(n + 1) + 1/6 F(n) F(n + 1) 3 6031 8 - 9/16 F(n + 1) + 1/30 F(n + 1) - ---- F(n) F(n + 1) 48 5825 7 2 1195 5 4 6 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - 1/2 F(n) F(n + 1) 48 48 4 3 683 4 5 193 3 6 + 1/3 F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 16 12 5 2309 5 + 7/8 F(n + 1) + ---- F(n) 48 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 963 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 9 124865 9 168 3 50 7 G(n) = ---- F(n + 1) - ------ F(n) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 8932 319 319 2 6 207584 8 162791 + 5/319 F(n) F(n + 1) - ------ F(n) F(n + 1) + ------ F(n) 2233 8932 8 20 8 40943 4 + 1/319 F(n) + --- F(n + 1) - ----- F(n) F(n + 1) 319 1276 60575 8 122 2 2 184006 2 3 + ----- F(n) F(n + 1) - --- F(n) F(n + 1) + ------ F(n) F(n + 1) 4466 319 2233 320785 2 7 238 3 24991 3 2 - ------ F(n) F(n + 1) + --- F(n) F(n + 1) + ----- F(n) F(n + 1) 4466 319 4466 38 3 5 807867 3 6 159 4 + --- F(n) F(n + 1) + ------ F(n) F(n + 1) - --- F(n) 319 8932 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 964 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 5 4 G(n) = -1/84 F(n) (30 F(n) F(n + 1) + 120350 F(n) F(n + 1) 5 2 2 4 5 + 72 F(n) F(n + 1) - 46 F(n) F(n + 1) - 18675 F(n) F(n + 1) 4 3 6 3 4 - 4064 F(n) F(n + 1) - 114975 F(n) F(n + 1) - 60 F(n) F(n + 1) 3 2 7 9 7 - 14 F(n + 1) + 49825 F(n) F(n + 1) - 25 F(n + 1) + 6 F(n + 1) 3 5 5 2 3 + 6 F(n) + 498 F(n) + 45 F(n + 1) - 49888 F(n) F(n + 1) 3 2 + 16915 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 965 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 4 145 2 2 G(n) = -5/22 F(n + 1) + 7 F(n) + 8/11 F(n) F(n + 1) + --- F(n) F(n + 1) 22 101 2 3 35 7 185 5 3 - --- F(n) F(n + 1) - -- F(n) F(n + 1) + --- F(n) F(n + 1) 22 22 11 213 6 2 32 3 5 4 5 - --- F(n) F(n + 1) + -- F(n) F(n + 1) - 3 F(n) F(n + 1) + 7/11 F(n) 11 11 7 3 2 8 106 8 - 3/2 F(n) F(n + 1) + 5 F(n) F(n + 1) + 5/22 F(n + 1) - --- F(n) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 966 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 28525 2 8 6056 3 2 405852 3 3 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) 22 319 319 237125 3 7 21019 4523 4 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 22 319 319 1307957 5 306975 9 3527 2 3 + ------- F(n) F(n + 1) - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 638 22 638 958985 2 4 1674703 6 61425 10 - ------ F(n) F(n + 1) - ------- F(n + 1) + ----- F(n + 1) 638 319 11 4289 463 2377 2 106642 2 - ---- F(n + 1) + --- F(n) - ---- F(n) - ------ F(n + 1) 638 319 319 319 4329 5 + ---- F(n + 1) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 967 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 360 2 133 1033 9 G(n) = ---- F(n + 1) + --- F(n) - ---- F(n + 1) + 2605/2 F(n + 1) 11 22 22 5 6 27617 5 - 75 F(n) F(n + 1) + 65/2 F(n + 1) - ----- F(n + 1) 22 3 3 45 2 2 4 + 80 F(n) F(n + 1) - -- F(n) - 15 F(n) F(n + 1) 22 3591 2 3 8 11915 4 - ---- F(n) F(n + 1) - 6495/2 F(n) F(n + 1) + ----- F(n) F(n + 1) 11 22 3 6 4735 3 2 117 + 5065/2 F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 22 11 2 7 + 280 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 968 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 22703 3 4 238647 6 G(n) = ----- F(n) F(n + 1) + 725/2 F(n) F(n + 1) + ------ F(n + 1) 935 1870 6 41212 9 162471 2 - 775/2 F(n) F(n + 1) + ----- F(n) F(n + 1) - ------ F(n + 1) 187 1870 535261 5 31480 9 409 2 - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - --- F(n) F(n + 1) 1870 187 22 2 5 32534 2 8 75751 10 - 25 F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n + 1) 187 1870 77279 10 46757 2 7 59 3 + ----- F(n) - ----- F(n) + 325/2 F(n + 1) + -- F(n) 1870 935 22 21973 2 4 1387 2 1790 3 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n + 1) 374 22 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 969 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 2 8 45965 2 4 G(n) = 4063/3 F(n) F(n + 1) + 5475/4 F(n) F(n + 1) - ----- F(n) F(n + 1) 28 2 2 123644 6 4 - 12/7 F(n) F(n + 1) - ------ F(n + 1) + 4/7 F(n + 1) 21 29549 2 2 74875 10 - ----- F(n + 1) - 52/7 F(n) + ----- F(n + 1) - 4/7 84 12 9 2845 3 - 46700/3 F(n) F(n + 1) + ---- F(n) F(n + 1) - 8/21 F(n) F(n + 1) 42 3 7 97271 5 3 + 36325/3 F(n) F(n + 1) + ----- F(n) F(n + 1) + 8/3 F(n) F(n + 1) 42 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 970 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 14888151 3 85929311 2 5 10311 G(n) = --------- F(n + 1) - -------- F(n) F(n + 1) - ----- 6061 12122 12122 7635225 11 155775 3 8931 4 + ------- F(n + 1) + ------ F(n) + ----- F(n + 1) 319 12122 12122 23139 3 5277 3 7575519 2 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + ------- F(n) F(n + 1) 6061 6061 12122 14474300 3 8 97299591 3 4 + -------- F(n) F(n + 1) + -------- F(n) F(n + 1) 319 12122 2098075 2 9 3063 2 2 718536 2 + ------- F(n) F(n + 1) - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 319 1102 6061 19179400 10 81400417 6 130180434 7 - -------- F(n) F(n + 1) + -------- F(n) F(n + 1) - --------- F(n + 1) 319 12122 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 971 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 306975 9 28525 2 8 962813 2 4 G(n) = ------- F(n) F(n + 1) + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 22 22 638 1577 2 3 2689 5 409680 3 3 + ---- F(n) F(n + 1) - ---- F(n + 1) + ------ F(n) F(n + 1) 638 638 319 7347 3 2 41719 107599 2 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - ------ F(n + 1) 638 638 319 61425 10 4435 2 669 2729 + ----- F(n + 1) - ---- F(n) - --- F(n) + ---- F(n + 1) 11 638 638 638 3133 4 1673746 6 1304767 5 + ---- F(n) F(n + 1) - ------- F(n + 1) + ------- F(n) F(n + 1) 319 319 638 237125 3 7 + ------ F(n) F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 972 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 8 2 4 G(n) = -1/84 F(n) (217175 F(n) F(n + 1) - 277873 F(n) F(n + 1) 3 3 3 3 7 + 130 F(n) F(n + 1) + 213343 F(n) F(n + 1) + 2169825 F(n) F(n + 1) 2 4 6 10 - 53313 F(n + 1) - 45 F(n + 1) - 1054069 F(n + 1) + 1107400 F(n + 1) 3 5 + 9774 F(n) F(n + 1) + 90 F(n) F(n + 1) + 423596 F(n) F(n + 1) 9 2 2 2 - 2754850 F(n) F(n + 1) - 175 F(n) F(n + 1) - 991 F(n) - 17) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 973 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 52413 5 2 532255 6 5 327 2 G(n) = ------ F(n) F(n + 1) + ------ F(n) F(n + 1) + --- F(n) 319 638 638 20 10 185 6 3572 7 20 11 + --- F(n + 1) + --- F(n) + ---- F(n) - --- F(n + 1) 319 638 319 319 838268 6 1587105 5 6 40 5 5 - ------ F(n) F(n + 1) - ------- F(n) F(n + 1) + --- F(n) F(n + 1) 1595 638 319 2715 5 70775 7 4 20 6 4 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) + --- F(n) F(n + 1) 319 22 319 40 9 720227 6 1200 5 - --- F(n) F(n + 1) + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 319 3190 319 1066357 2 1181 843145 2 5 - ------- F(n) F(n + 1) + ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 3190 319 638 2590 2 4 695203 2 34695 10 + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 319 1595 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 974 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(4 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 657 4 2364 153 11799 2 7 G(n) = ---- F(n) F(n + 1) - ---- F(n) + ---- F(n + 1) - ----- F(n) F(n + 1) 4147 4147 4147 8294 401371 5 647331 9 8850428 6 4 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - ------- F(n) F(n + 1) 65395 26158 65395 3105 8 302417 2 2200273 6 + ---- F(n) F(n + 1) - ------ F(n + 1) + ------- F(n + 1) 4147 65395 130790 1671 5 786957 10 1971 10 3105 9 - ---- F(n + 1) - ------ F(n + 1) - ---- F(n) + ---- F(n) 8294 65395 319 4147 18663 2 3 15525 8 9113759 8 2 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ------- F(n) F(n + 1) 8294 8294 130790 5240273 9 10231576 7 3 269299 + ------- F(n) F(n + 1) + -------- F(n) F(n + 1) - ------ F(n) F(n + 1) 130790 65395 65395 3726 7 2 + ---- F(n) F(n + 1) 4147 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 975 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 8 2 2 G(n) = 1/130200 F(n) (-82150 F(n) - 82150 F(n + 1) - 90520 F(n) F(n + 1) 2 9 3 - 28314482 F(n + 1) + 4256850 F(n) F(n + 1) + 168485 F(n) F(n + 1) 9 2 6 7 + 86666090 F(n) F(n + 1) - 98580 F(n) F(n + 1) + 205375 F(n) F(n + 1) 10 3 - 1537182 F(n + 1) + 1988522 F(n) F(n + 1) - 16275 F(n) F(n + 1) 5 7 - 59781372 F(n) F(n + 1) - 156085 F(n) F(n + 1) 7 3 8 2 - 128442690 F(n) F(n + 1) + 93756400 F(n) F(n + 1) 6 4 10 2 + 29845464 F(n + 1) + 69750 F(n + 1) + 26037878 F(n) - 24393328 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 976 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(4 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 78373250 11 9243 5 31055125 12 G(n) = --------- F(n) F(n + 1) + ----- F(n + 1) + -------- F(n + 1) 319 12122 319 10623 300 186206059 4 - ----- F(n + 1) - ---- F(n) - --------- F(n + 1) 12122 6061 12122 3273535 7 6456 4 27418621 3 + ------- F(n) F(n + 1) - ---- F(n) F(n + 1) + -------- F(n) F(n + 1) 209 6061 6061 1401 2 3 6812017 2 2 - ---- F(n) F(n + 1) - ------- F(n) F(n + 1) 418 6061 57919375 3 9 549930485 3 5 + -------- F(n) F(n + 1) + --------- F(n) F(n + 1) 319 12122 20217 3 2 1394442 3 + ----- F(n) F(n + 1) + ------- F(n) F(n + 1) 6061 6061 9711125 2 10 187779700 2 6 + ------- F(n) F(n + 1) - --------- F(n) F(n + 1) 319 6061 993669715 8 108798 - --------- F(n + 1) - ------ 12122 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 977 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 9397035 2 9 1200 3 3 971577 3 4 G(n) = ------- F(n) F(n + 1) + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) 319 319 319 2880 4 2 1550939 4 3 105 4 6 - ---- F(n) F(n + 1) - ------- F(n) F(n + 1) - --- F(n) F(n + 1) 319 638 638 7344560 4 7 81 396125 2 - ------- F(n) F(n + 1) + -- F(n) F(n + 1) + ------ F(n) F(n + 1) 319 29 638 765 5 3572516 6 105 9 - --- F(n) F(n + 1) + ------- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 3770830 10 2583 2 1575 2 4 - ------- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 22 319 49011 2 5 105 2 8 105 10 - ----- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n + 1) 11 638 638 1586815 3 8 387 2 8043 3 105 3 7 - ------- F(n) F(n + 1) - --- F(n) + ---- F(n) - --- F(n) F(n + 1) 638 638 638 319 105 11 + --- F(n + 1) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 978 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(4 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 9 G(n) = -1/84 F(n) (-47 F(n) + 1163750 F(n) F(n + 1) + 304 F(n + 1) 3 5 2 - 397892 F(n + 1) - 280 F(n + 1) + 95718 F(n) F(n + 1) 4 6 10 + 595 F(n) F(n + 1) + 1440415 F(n) F(n + 1) - 11568375 F(n) F(n + 1) 2 3 2 3 4 - 16682 F(n) F(n + 1) - 1015 F(n) F(n + 1) + 1352305 F(n) F(n + 1) 3 8 2 3 2 5 + 8840125 F(n) F(n + 1) + 490 F(n) F(n + 1) - 1308850 F(n) F(n + 1) 7 11 3 - 4220370 F(n + 1) + 4618250 F(n + 1) + 1559 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 979 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(4 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 126375 9 2 6165 2 32931 2 G(n) = ------- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 8294 4147 8294 3177 2 5 124965 5 6 127665 6 - ---- F(n) F(n + 1) + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 319 4147 8294 13857 10 23817 10 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 754 4147 184108051 2 2 417486823 11 - --------- F(n) F(n + 1) - --------- F(n) F(n + 1) 32857 65714 20949839 7 303492381 11 - -------- F(n) F(n + 1) + --------- F(n) F(n + 1) 2266 32857 1243074403 10 2 494827987 3 + ---------- F(n) F(n + 1) - --------- F(n) F(n + 1) 65714 65714 505699860 9 3 535971104 2 6 + --------- F(n) F(n + 1) - --------- F(n) F(n + 1) 32857 32857 98781732 8 49914160 98593865 12 231264843 12 + -------- F(n + 1) - -------- + -------- F(n) + --------- F(n + 1) 32857 32857 65714 65714 26478 3 36843 11 164494586 4 501 11 + ----- F(n + 1) + ----- F(n + 1) - --------- F(n + 1) + --- F(n) 4147 4147 32857 638 11637 7 - ----- F(n + 1) 754 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 980 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 4 3 7 G(n) = -1/30 F(n) (-F(n) - 39 F(n) - 1550 F(n) F(n + 1) - 59 F(n + 1) 4 3 4 4 + 180 F(n) F(n + 1) + 915 F(n) F(n + 1) - 90 F(n) F(n + 1) 5 2 3 6 + 842 F(n) F(n + 1) + 49 F(n + 1) - 470 F(n) F(n + 1) 2 5 7 + 133 F(n) F(n + 1) - 58 F(n + 1) + 48 F(n + 1) + 100 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 981 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(n + j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 743 5 9 325 5 4 G(n) = --- F(n) F(n + 1) + 5/22 F(n + 1) + --- F(n) F(n + 1) 110 22 1385 7 2 284 157 4 265 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) 44 55 22 44 211 2 65 5 6 151 6 208 2 3 - --- F(n) - -- F(n) - 5/22 F(n + 1) + --- F(n) + --- F(n) F(n + 1) 110 44 110 11 2 4 138 5 395 6 3 - 14 F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 11 22 955 8 35 8 - --- F(n) F(n + 1) - -- F(n) F(n + 1) 22 44 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 982 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(n + j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 8 G(n) = -1/6 F(n) (-34 + 13080 F(n) F(n + 1) + 6135 F(n + 1) 7 6 2 2 - 14865 F(n) F(n + 1) - 195 F(n + 1) - 957 F(n) F(n + 1) 5 3 3 + 450 F(n) F(n + 1) - 59 F(n) F(n + 1) - 480 F(n) F(n + 1) 2 6 4 3 - 105 F(n) F(n + 1) - 6104 F(n + 1) + 2533 F(n) F(n + 1) 3 2 4 2 2 + 307 F(n) F(n + 1) + 90 F(n) F(n + 1) + 10 F(n) + 194 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 983 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 79 2 110889 6 1537 G(n) = 205/2 F(n + 1) - -- F(n) - ------ F(n + 1) + ---- F(n) F(n + 1) 11 22 22 6 21555 5 14942 3 3 - 240 F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 11 11 33907 2 4 211 2 9 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) - 13550 F(n) F(n + 1) 22 22 3 7 3 4 402 2 + 10275 F(n) F(n + 1) + 475/2 F(n) F(n + 1) + --- F(n) F(n + 1) 11 13 3 3953 2 1130 3 2 5 + -- F(n) - ---- F(n + 1) - ---- F(n + 1) - 55/2 F(n) F(n + 1) 11 11 11 2 8 10 + 2875/2 F(n) F(n + 1) + 5400 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 984 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 4 4 4 6 G(n) = -575/2 F(n) F(n + 1) + 6 F(n) F(n + 1) + 628/3 F(n) F(n + 1) 5 5 3 5 5 + 421/6 F(n) F(n + 1) - 128/5 F(n) F(n + 1) - 2933/6 F(n) F(n + 1) 6 2 6 4 7 + 30 F(n) F(n + 1) - 1123/3 F(n) F(n + 1) - 31/2 F(n) F(n + 1) 7 3 3 4 + 5147/6 F(n) F(n + 1) - 12/5 F(n) F(n + 1) + 9/5 F(n + 1) 2 6 10 6 4 - 5/6 F(n + 1) + 4/3 F(n + 1) - 1/2 F(n + 1) - 9 F(n) + 9/2 F(n) 3 7 8 + 76/3 F(n) F(n + 1) - 3/2 - 3/10 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 985 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(n + 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 636550 10 1277 2 101 6 G(n) = ------- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n + 1) 11 11 22 5040247 3 4 73 2 4 2197586 2 5 + ------- F(n) F(n + 1) - -- F(n) F(n + 1) - ------- F(n) F(n + 1) 638 22 319 958325 3 8 709 197149 2 + ------ F(n) F(n + 1) + --- F(n) F(n + 1) + ------ F(n) F(n + 1) 22 638 319 107 5 4051129 6 2889 2 - --- F(n) F(n + 1) + ------- F(n) F(n + 1) - ---- F(n + 1) 11 638 638 13141371 7 3891 3 63 2 506525 11 - -------- F(n + 1) + ---- F(n) - --- F(n) + ------ F(n + 1) 638 319 319 22 70775 2 9 243 3 3 773947 3 + ----- F(n) F(n + 1) + --- F(n) F(n + 1) - ------ F(n + 1) 11 22 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 986 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(n + 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 279 3 4 769 2 4 G(n) = -149/5 F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 22 110 2 8 151 3 7 277 6 + 77/5 F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 11 22 339 2 5 6939 7 3 202 5 2 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 44 55 11 398 4 6 15 2 23 7 3 - --- F(n) F(n + 1) + -- F(n + 1) + -- F(n + 1) - 3/4 F(n + 1) 11 22 44 541 4 3 24 3 469 9 6 - --- F(n) F(n + 1) + -- F(n) + --- F(n) F(n + 1) - 5/11 F(n + 1) 44 11 11 1935 8 2 112 10 - ---- F(n) F(n + 1) - --- F(n) 22 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 987 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 5 7 1699 8 3 G(n) = 103/2 F(n) F(n + 1) + 115/2 F(n) F(n + 1) - ---- F(n) F(n + 1) 11 3691 9 2 2592 10 4 4 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - 82 F(n) F(n + 1) 11 11 19 3 4 493 3 5 301 3 8 - -- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 22 10 22 101 2 9 3 19 2 + --- F(n) F(n + 1) - 114/5 F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 3 7 361 3 91 4 + 4 F(n) F(n + 1) - 7/2 F(n) F(n + 1) - --- F(n) - -- F(n) 22 10 1213 7 8 1741 11 11 + ---- F(n) + 33/5 F(n) - ---- F(n) - 5/22 + 5/22 F(n + 1) 11 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 988 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 8 G(n) = -1/84 F(n) (8840125 F(n) F(n + 1) + 248 F(n + 1) 2 9 3 2 3 4 + 1163750 F(n) F(n + 1) - 539 F(n) F(n + 1) + 1352305 F(n) F(n + 1) 2 2 3 2 5 - 16696 F(n) F(n + 1) + 14 F(n) F(n + 1) - 1308850 F(n) F(n + 1) 4 6 10 + 707 F(n) F(n + 1) + 1440415 F(n) F(n + 1) - 11568375 F(n) F(n + 1) 7 11 2 - 4220370 F(n + 1) + 4618250 F(n + 1) + 95676 F(n) F(n + 1) 3 5 3 - 397864 F(n + 1) - 308 F(n + 1) + 1573 F(n) - 61 F(n)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 989 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(n + 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 12 526 3 20 11 6268 4 G(n) = ---- F(n + 1) + --- F(n) + --- F(n + 1) - ---- F(n) 319 319 319 319 28416195 7 5550 6 1446436 3 + -------- F(n) F(n + 1) - ---- F(n) F(n + 1) + ------- F(n) F(n + 1) 638 319 319 5602 2 40 3 8 4846720 4 4 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) - ------- F(n) F(n + 1) 319 319 319 63457 3 39266020 2 10 + ----- F(n) F(n + 1) + -------- F(n) F(n + 1) 319 319 4529815 3 5 30 3 4 26495 2 5 + ------- F(n) F(n + 1) + -- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 29 638 354461 2 2 2162 2 31309045 11 - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) - -------- F(n) F(n + 1) 319 319 638 40 10 20 4 7 24175 4 3 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 638 4114415 3 9 59656665 4 8 20 2 9 - ------- F(n) F(n + 1) - -------- F(n) F(n + 1) - --- F(n) F(n + 1) 319 638 319 9418025 2 6 - ------- F(n) F(n + 1) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 990 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 6 8 2 G(n) = 3/2 - 15/2 F(n) F(n + 1) - 335 F(n + 1) - 19/2 F(n) 3 3 5 - 16 F(n) F(n + 1) - 142 F(n) F(n + 1) + 20989/6 F(n) F(n + 1) 9 7 + 541/6 F(n) F(n + 1) - 23075 F(n) F(n + 1) + 815 F(n) F(n + 1) 2 4 2 2 3 5 - 4761/2 F(n) F(n + 1) + 54 F(n) F(n + 1) - 705 F(n) F(n + 1) 3 3 2 8 3 7 + 5672/3 F(n) F(n + 1) + 1925 F(n) F(n + 1) + 36125/2 F(n) F(n + 1) 6 2 4 - 26344/3 F(n + 1) - 2887/6 F(n + 1) + 667/2 F(n + 1) 10 + 18525/2 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 991 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(2 n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 211591 7 11031 9 2 1634 7 G(n) = ------ F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 7700 77 21 1221271 10 2 6 - ------- F(n) F(n + 1) - 636/7 F(n) F(n + 1) 15400 1370253 8 3 65 2 2 6529 2 - ------- F(n) F(n + 1) - -- F(n) F(n + 1) - ----- F(n) F(n + 1) 3080 21 15400 602493 2 5 80202 2 162663 10 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 3080 1925 1540 1630 7 16257 6 3 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - 514/7 F(n) F(n + 1) 21 700 4 873141 11 357 11 230 8 7013 + 63/2 F(n + 1) - ------ F(n + 1) + --- F(n) + --- F(n) - ---- 15400 22 21 462 453459 3 695 8 + ------ F(n + 1) - --- F(n + 1) 15400 42 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 992 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(2 n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 11 3 G(n) = 76975/2 F(n + 1) + 61/2 F(n + 1) - 19/6 F(n) - 3537 F(n + 1) 9 5 7 - 1475 F(n + 1) + 2889/2 F(n + 1) - 69901/2 F(n + 1) 8 6 4 + 7275/2 F(n) F(n + 1) + 69587/6 F(n) F(n + 1) - 1247/2 F(n) F(n + 1) 2 2 3 8 + 1733/2 F(n) F(n + 1) - 464/3 F(n) F(n + 1) + 73400 F(n) F(n + 1) 3 6 3 4 3 2 - 2975 F(n) F(n + 1) + 71131/6 F(n) F(n + 1) - 327/2 F(n) F(n + 1) 2 7 2 3 10 - 375/2 F(n) F(n + 1) + 312 F(n) F(n + 1) - 193025/2 F(n) F(n + 1) 2 5 2 9 3 - 66611/6 F(n) F(n + 1) + 20075/2 F(n) F(n + 1) + 91/6 F(n) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 993 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 n - j) F(2 n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 1842 3 2 5275 3 G(n) = -- F(n) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 11 22 536225 2 10 5100 2 7 507465 2 6 + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 22 11 22 10273 2 3 14310 2 2 47 3 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 22 61200 4 8 1800 4 5 141275 4 4 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 11 11 22 3 9 3 6 101775 3 5 - 60875 F(n) F(n + 1) - 1075 F(n) F(n + 1) + ------ F(n) F(n + 1) 11 24465 5 4 56515 5 3 518380 5 7 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 22 11 11 1013 4 12 79 5 515 4 - ---- F(n) F(n + 1) + 5/22 F(n + 1) + -- F(n) - --- F(n) 22 22 22 9 - 5/22 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 994 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 n - j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 10 6 8 G(n) = -12575/2 F(n + 1) + 5976 F(n + 1) - 403630/3 F(n + 1) 12 2 + 157250 F(n + 1) - 175/3 F(n) F(n + 1) + 37/6 F(n) 2 10 3 5 + 94875/2 F(n) F(n + 1) + 136325/2 F(n) F(n + 1) 3 7 3 3 2 6 - 12300 F(n) F(n + 1) - 1237 F(n) F(n + 1) - 147170/3 F(n) F(n + 1) 2 8 2 2 2 4 - 2525/2 F(n) F(n + 1) - 5681/4 F(n) F(n + 1) + 3179/2 F(n) F(n + 1) 9 11 7 + 15650 F(n) F(n + 1) - 792625/2 F(n) F(n + 1) + 89455/3 F(n) F(n + 1) 5 3 3 9 - 2394 F(n) F(n + 1) + 6846 F(n) F(n + 1) + 589375/2 F(n) F(n + 1) 1585 4 2 4 + ---- - 625/4 F(n) + 623/2 F(n + 1) - 91355/4 F(n + 1) 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 995 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 G(n) = - 5/22 + 5/22 F(n + 1) - 3/22 F(n) + 9/22 F(n) F(n + 1) 3 2 31 4 15 5 - 5/22 F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) 22 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 996 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 3 G(n) = -1/4 F(n) (8 F(n) - 63 F(n) F(n + 1) - 18 F(n) F(n + 1) 3 2 5 + 75 F(n) F(n + 1) - 30 F(n + 1) + 28 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 997 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4393 2 1946 3 4 12082 6 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 319 28763 2 5 27241 4 3 23995 2 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 638 105 2019 3 105 3 - --- + ---- F(n) + --- F(n + 1) 638 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 998 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 2 211 2 427 6 G(n) = --- F(n) - 2/11 F(n) - 5/22 F(n + 1) - --- F(n + 1) + --- F(n + 1) 11 11 22 456 3 3 1029 5 + 8 F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 22 15 2 4 - -- F(n) F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 999 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 6 2 6 G(n) = 1/20 F(n) (-5 F(n) + 65 F(n) - 36 F(n + 1) + 51 F(n + 1) 2 2 3 - 7 F(n) F(n + 1) + 5 F(n) F(n + 1) - 10 F(n) F(n + 1) 4 2 5 4 + 185 F(n) F(n + 1) - 143 F(n) F(n + 1) - 5 F(n + 1) 3 5 + 10 F(n) F(n + 1) - 110 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1000 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 173 2 5 4671 6 G(n) = --- F(n) F(n + 1) - ---- F(n) F(n + 1) - 4/11 F(n) F(n + 1) 22 22 67 3 434 2 3887 3 4 + -- F(n) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 22 11 22 315 2 952 7 2 2 - --- F(n) F(n + 1) + --- F(n + 1) - 1/22 F(n) - 5/22 F(n + 1) 22 11 1899 3 - ---- F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1001 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 2 G(n) = ) F(j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 4 2 64 5 G(n) = 7/6 F(n) F(n + 1) - 4 F(n) F(n + 1) + -- F(n) F(n + 1) 15 13 6 2 6 - -- F(n) F(n + 1) - F(n) + 9/20 F(n + 1) - 9/20 F(n + 1) 30 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1002 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 67 4 3 57 3 618 3 153 2 5 G(n) = ---- F(n) F(n + 1) - --- F(n + 1) + --- F(n) + --- F(n) F(n + 1) 638 638 319 638 17 7 20 3552 6 625 3 4 + --- F(n + 1) + --- - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 319 319 3516 5 2 + ---- F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1003 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 6 29 6 G(n) = 7/55 F(n + 1) + 1/10 F(n + 1) + 7/22 F(n) - 5/22 F(n + 1) - -- F(n) 22 24 4 2 221 5 36 - -- F(n) F(n + 1) + --- F(n) F(n + 1) - -- F(n) F(n + 1) 11 55 55 3 3 - 2/11 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1004 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 2 4 G(n) = -1/12 F(n) (F(n + 1) - 2 F(n) F(n + 1) - 25 F(n) + F(n) 3 2 2 2 2 4 + 2 F(n) F(n + 1) - 402 F(n + 1) - F(n) F(n + 1) - 139 F(n) F(n + 1) 3 3 6 + 969 F(n) F(n + 1) + 403 F(n + 1) + 130 F(n) F(n + 1) 5 - 937 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1005 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 91 20 8315 7 846 4 G(n) = ---- F(n) + --- F(n + 1) + ---- F(n) F(n + 1) - --- F(n) 319 319 638 319 2790 3 5 20 21971 7 1021 2 2 + ---- F(n) F(n + 1) - --- + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 638 319 9426 2 6 9003 3 8179 3 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1006 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2077 6 725 2 1784 3 4 G(n) = ----- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 22 11 249 2 26 2 5 + 7/11 F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 1701 3 28 3 2 2 853 7 - ---- F(n + 1) + -- F(n) - 6/11 F(n) - 5/22 F(n + 1) + --- F(n + 1) 22 11 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1007 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 5 G(n) = -1/12 F(n) (-2 F(n + 1) + F(n)) (37 F(n) - 90 F(n) F(n + 1) 4 2 4 3 3 3 + 92 F(n) F(n + 1) - F(n) - 52 F(n) F(n + 1) - 2 F(n) F(n + 1) 2 2 2 4 3 4 + F(n) F(n + 1) + 14 F(n) F(n + 1) + 2 F(n) F(n + 1) - F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1008 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 281 3 63 8 8 31 3 G(n) = ---- F(n) F(n + 1) + -- F(n + 1) - 41/8 F(n) - -- F(n + 1) 176 88 44 101 3 4 111 21 7 177 7 - --- F(n) - 7/4 F(n + 1) + --- + -- F(n + 1) + --- F(n) 88 88 44 88 381 5 2 4087 5 3 1605 6 2 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 88 176 88 1745 7 465 3 4 1471 3 5 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 176 88 176 177 2 5 105 6 - --- F(n) F(n + 1) + --- F(n) F(n + 1) 44 44 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1009 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 8453 3 6 690 37056 5 759 2 7 G(n) = ------ F(n) F(n + 1) - ---- + ----- F(n) + ---- F(n) F(n + 1) 12122 6061 6061 1102 343661 4 5 1515245 7 2 23322 6 3 + ------ F(n) F(n + 1) + ------- F(n) F(n + 1) - ----- F(n) F(n + 1) 12122 12122 319 53 317 5 299511 5 4 + --- F(n + 1) - ---- F(n + 1) - ------ F(n) F(n + 1) 319 6061 6061 20348 4 - ----- F(n) F(n + 1) 551 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1010 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 6 5 G(n) = 1/12 F(n) (F(n) + 23 F(n) - 403 F(n + 1) + 937 F(n) F(n + 1) 3 3 4 3 - 969 F(n) F(n + 1) + F(n + 1) - 136 F(n) F(n + 1) - 2 F(n) F(n + 1) 2 2 2 4 3 2 - F(n) F(n + 1) + 139 F(n) F(n + 1) + 2 F(n) F(n + 1) + 408 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1011 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2790 3 5 20 2649 4 137 20 G(n) = ---- F(n) F(n + 1) - --- - ---- F(n) + --- F(n) + --- F(n + 1) 319 319 638 638 319 40943 2 2 6484 3 101357 3 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) 638 319 638 50770 7 122400 2 6 109855 4 4 - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 319 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1012 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 2 G(n) = 1/6 F(n) F(n + 1) - 419/6 F(n) F(n + 1) - 1/12 F(n) 395 2 6 47 4283 4 8 + --- F(n) F(n + 1) - -- - ---- F(n + 1) + 2165/6 F(n + 1) 12 12 12 3 3 5 3 + 57/2 F(n) F(n + 1) + 1475/2 F(n) F(n + 1) + 470/3 F(n) F(n + 1) 7 - 5315/6 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1013 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 2 20 9 97 2 137 G(n) = --- F(n + 1) - --- F(n + 1) - --- F(n) + --- F(n) F(n + 1) 319 319 638 319 469895 4 5 17565 3 6 41752 3 2 - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 319 560070 2 7 228460 8 3925 + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) + ---- F(n) 319 319 638 17887 2 3 15992 4 15615 4 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 58 319 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1014 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 2 6 3 3 G(n) = 1/24 F(n) (16 F(n) F(n + 1) + 122 F(n) F(n + 1) - 2 F(n) F(n + 1) 4 5 8 6 + 6 F(n + 1) - 6 F(n) F(n + 1) + 140 F(n) + 4 F(n) 3 5 4 2 5 - 35 F(n) F(n + 1) - 16 F(n) F(n + 1) + 2 F(n) F(n + 1) 5 3 6 2 3 - 1645 F(n) F(n + 1) + 2050 F(n) F(n + 1) - 469 F(n) F(n + 1) 3 2 7 - 12 F(n) F(n + 1) + 6 F(n + 1) - 161 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1015 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 2 G(n) = ) F(j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 5 5 4 G(n) = -1/616 F(n) (-96800 F(n) F(n + 1) + 926600 F(n) F(n + 1) 4 2 3 2 7 - 40707 F(n) F(n + 1) - 401697 F(n) F(n + 1) + 400400 F(n) F(n + 1) 3 2 5 5 + 148456 F(n) F(n + 1) - 51 F(n + 1) + 5544 F(n) 3 6 - 942000 F(n) F(n + 1) + 255 F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1016 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 105 21 105 17891 7 G(n) = --- + --- F(n) - --- F(n + 1) + ----- F(n) F(n + 1) 638 638 638 638 8295 3 9409 3 4749 2 2 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 638 638 10446 2 6 1020 8 4830 3 5 - ----- F(n) F(n + 1) - ---- F(n) + ---- F(n) F(n + 1) 319 319 319 8315 7 + ---- F(n) F(n + 1) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1017 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 690 9 1857 18459607 5 G(n) = ----- F(n + 1) + ---- F(n) + -------- F(n) F(n + 1) 6061 6061 6061 1380 4 93965295 2 8 690 2 7 + ---- F(n) F(n + 1) + -------- F(n) F(n + 1) + ---- F(n) F(n + 1) 6061 12122 6061 724006 2 4 690 2 3 18972255 9 - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) - -------- F(n) F(n + 1) 551 6061 6061 1380 8 5063667 4 2 690 4 + ---- F(n) F(n + 1) - ------- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 12122 6061 3039020 3 7 1380 3 6 4333407 3 3 - ------- F(n) F(n + 1) - ---- F(n) F(n + 1) + ------- F(n) F(n + 1) 6061 6061 6061 1380 3 2 37721335 4 6 690 4 5 - ---- F(n) F(n + 1) - -------- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 6061 6061 56406 2 46127 690 10 - ----- F(n) + ----- F(n) F(n + 1) + ---- F(n + 1) 6061 551 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1018 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 73330 3 2 1168250 8 20821 2 3 G(n) = ----- F(n) F(n + 1) - ------- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 58 21 390201 4 90175 2 7 + --- F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 319 638 319 63 2 105 2 469875 9 1977 2847 - --- F(n) - --- F(n + 1) + ------ F(n + 1) + ---- F(n) - ---- F(n + 1) 319 638 319 319 58 454164 5 922225 3 6 - ------ F(n + 1) + ------ F(n) F(n + 1) 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1019 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 488250 2 8 1962075 10 105 3 147 3 G(n) = ------ F(n) F(n + 1) + ------- F(n + 1) - --- F(n + 1) + --- F(n) 319 319 638 638 3760700 3 7 494854 3 3 99861 2 4 + ------- F(n) F(n + 1) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 58 189 2 4912900 9 694394 5 - --- F(n) F(n + 1) - ------- F(n) F(n + 1) + ------ F(n) F(n + 1) 319 319 319 26966 5889 2 63 2 + ----- F(n) F(n + 1) - ---- F(n) + --- F(n) F(n + 1) 319 638 638 3656771 6 133637 2 - ------- F(n + 1) - ------ F(n + 1) 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1020 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 249 2 290 3 4 6 G(n) = ---- F(n) F(n + 1) - --- F(n) F(n + 1) - 3/11 F(n) F(n + 1) 22 11 117 4 3 2 5 - --- F(n) F(n + 1) + 13 F(n) F(n + 1) + 3/22 F(n) F(n + 1) 11 2 2 3 45 3 - 5/22 F(n + 1) - 1/22 F(n) + 5/22 F(n + 1) + -- F(n) 22 368 5 2 + --- F(n) F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1021 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 5 3 4 G(n) = -1/12 F(n) (-190 F(n) F(n + 1) + 4330 F(n) F(n + 1) - F(n) 3 3 6 + 37 F(n) + 2 F(n + 1) - 1978 F(n + 1) - 4725 F(n) F(n + 1) 2 7 2 + 797 F(n) F(n + 1) + 1970 F(n + 1) - 242 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1022 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 120 7 8 3 G(n) = -5/22 F(n + 1) + --- F(n) F(n + 1) + 5/22 - 17 F(n) + 4/11 F(n) 11 295 4 2 116 3 + --- F(n) - 1/22 F(n) F(n + 1) - --- F(n) F(n + 1) 22 11 409 3 5 435 2 2 435 2 6 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 11 22 11 158 3 2 - --- F(n) F(n + 1) - 3/22 F(n) F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1023 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 2 G(n) = ) F(j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 G(n) = -1/12 F(n) (2 F(n + 1) - 1970 F(n + 1) + 49 F(n) - F(n) 3 4 2 6 + 4330 F(n) F(n + 1) - 250 F(n) F(n + 1) - 4725 F(n) F(n + 1) 2 7 2 5 + 785 F(n) F(n + 1) + 1970 F(n + 1) - 190 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1024 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 6 469895 4 5 40 5 G(n) = --- F(n + 1) - ------ F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 319 45 97 2 20 9 228460 8 - --- F(n) F(n + 1) - --- F(n) - --- F(n + 1) - ------ F(n) F(n + 1) 638 638 319 319 20 2 4 20 4 2 83823 3 2 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 638 560070 2 7 7813 4 15673 4 + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 11 319 17565 3 6 40 3 3 8987 2 3 - ----- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 29 3925 + ---- F(n) 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1025 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 85 500 2 6 2 G(n) = 340 F(n + 1) - -- + --- F(n) F(n + 1) + 5/11 F(n) F(n + 1) 22 11 645 3 7515 3 5 9230 7 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 22 11 11 3 3 3317 3 2 - 5/22 F(n + 1) - 3/22 F(n) + ---- F(n) F(n + 1) - 7/11 F(n) F(n + 1) 22 1525 2 2 3695 4 - ---- F(n) F(n + 1) - ---- F(n + 1) 22 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1026 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 3 7 2 G(n) = -1/8 F(n + 1) + 1/8 F(n + 1) + 371/8 F(n) F(n + 1) 2431 8 1963 4 5 461 5 4 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 96 96 12 6 2423 6 3 131 3 6 - 1/12 F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 48 24 5 9 9 7 - 3/32 F(n + 1) + 7/48 F(n + 1) - 5/96 F(n + 1) + 37/6 F(n) - 1/6 F(n) 5 2 2 5 3 4 + 5/12 F(n) F(n + 1) + 7/8 F(n) F(n + 1) - 2/3 F(n) F(n + 1) 13 4 3 - -- F(n) F(n + 1) 24 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1027 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 775 9 453 963 4 G(n) = --- F(n + 1) - --- F(n) + --- F(n + 1) - 7/22 F(n + 1) 22 11 22 1733 5 31025 4 5 3 - ---- F(n + 1) - ----- F(n) F(n + 1) + 1/11 + 3/11 F(n) F(n + 1) 22 22 518 5 9206 4 21 3 + --- F(n) + ---- F(n) F(n + 1) + -- F(n) F(n + 1) 11 11 22 37575 2 7 37 2 2 8575 8 + ----- F(n) F(n + 1) - -- F(n) F(n + 1) - ---- F(n) F(n + 1) 22 22 11 3966 2 3 - ---- F(n) F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1028 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 2 7 G(n) = 1/60 F(n) (5 F(n) F(n + 1) + 345 F(n) - 20 F(n) - 134 F(n) F(n + 1) 6 2 4 4 7 + 2172 F(n) F(n + 1) + 2685 F(n) F(n + 1) - 1605 F(n) F(n + 1) 5 3 3 2 2 - 4190 F(n) F(n + 1) + 134 F(n) F(n + 1) + 578 F(n) F(n + 1) 2 - 5 F(n + 1) + 35) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1029 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 227435 3 3 988710 9 961813 5 G(n) = ------ F(n) F(n + 1) - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 319 319 319 26908 1962095 4 6 267419 4 2 + ----- F(n) F(n + 1) - ------- F(n) F(n + 1) - ------ F(n) F(n + 1) 319 319 638 163490 3 7 2450345 2 8 - ------ F(n) F(n + 1) + ------- F(n) F(n + 1) 319 319 415526 2 4 291 2 20 3 - ------ F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n + 1) 319 638 319 20 10 85 3 5657 2 46 2 - --- F(n + 1) - --- F(n) - ---- F(n) + --- F(n) F(n + 1) 319 638 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1030 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 9 3 6 3 G(n) = -1/60 F(n) (1725 F(n) - 25 F(n) - 1160 F(n) - 16728 F(n) F(n + 1) 2 3 8 2 - 232 F(n) F(n + 1) + 4961 F(n) F(n + 1) + 50 F(n) F(n + 1) 4 5 9 + 1254 F(n) F(n + 1) + 2460 F(n + 1) - 2490 F(n + 1) 7 2 8 + 8425 F(n) F(n + 1) + 1760 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1031 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 108 6 2 4189 6 3 53 2 2 G(n) = --- F(n) F(n + 1) - ---- F(n) F(n + 1) - -- F(n) F(n + 1) 55 110 55 173 5 4 59 9 9 8 + --- F(n) F(n + 1) - 5/22 + -- F(n) + 5/22 F(n + 1) - 2/11 F(n) 11 11 5 3 7 947 7 2 + 4/11 F(n) F(n + 1) + 6/11 F(n) F(n + 1) + --- F(n) F(n + 1) 22 324 8 54 7 67 8 - --- F(n) F(n + 1) + -- F(n) F(n + 1) - --- F(n) F(n + 1) 11 55 110 519 2 3 2 6 64 3 + --- F(n) F(n + 1) - 2 F(n) F(n + 1) - -- F(n) F(n + 1) 110 55 13 4 23 - -- F(n) F(n + 1) + -- F(n) 10 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1032 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 2 G(n) = ) F(j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 7 78275 10 G(n) = 1/6 F(n) F(n + 1) + 37700/3 F(n) F(n + 1) + ----- F(n + 1) 12 2 5107 2 - 37/4 F(n) - ---- F(n + 1) + 1/4 + 256/3 F(n) F(n + 1) 12 3 3 3 4 + 4751/3 F(n) F(n + 1) - 7/6 F(n) F(n + 1) - 1/4 F(n + 1) 6 5 2 4 - 18292/3 F(n + 1) + 13957/6 F(n) F(n + 1) - 7151/4 F(n) F(n + 1) 2 2 9 2 8 + 3/4 F(n) F(n + 1) - 48925/3 F(n) F(n + 1) + 6175/4 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1033 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 79 3 2 343 3 3 1640 3 7 G(n) = -- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 44 11 11 12 4 1585 4 2 1323 4 6 - -- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 22 22 12 8 8 2 537 9 - -- F(n) F(n + 1) - 170 F(n) F(n + 1) + --- F(n) F(n + 1) 11 11 24 2 3 696 2 4 - 8/11 F(n) F(n + 1) - -- F(n) F(n + 1) + --- F(n) F(n + 1) 11 11 670 2 8 215 10 49 3 6 - --- F(n) F(n + 1) - --- F(n) - -- F(n) F(n + 1) 11 22 44 23 4 15 2 7 59 5 6 + -- F(n) F(n + 1) + -- F(n) F(n + 1) + -- F(n) + 5/22 F(n + 1) 22 11 44 9 25 9 - 5/22 F(n + 1) - -- F(n) 44 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1034 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 22427 7 18485 2 185 453 3 G(n) = ----- F(n + 1) + ----- F(n) F(n + 1) + --- + --- F(n) 319 638 638 638 2711 2 1436 2 5 45039 3 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n + 1) 319 319 638 53978 6 48656 3 4 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1035 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 122115 2 7 1040778 3 2 G(n) = ------ F(n) F(n + 1) + ------- F(n) F(n + 1) 551 6061 3059445 3 6 173358 2 3 194504 + ------- F(n) F(n + 1) - ------ F(n) F(n + 1) - ------ F(n + 1) 1102 551 6061 35751 8263511 5 615 3807685 8 + ----- F(n) - ------- F(n + 1) + ----- - ------- F(n) F(n + 1) 12122 6061 12122 1102 7240679 4 80935 9 + ------- F(n) F(n + 1) + ----- F(n + 1) 12122 58 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1036 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 2 2 5 G(n) = 299/4 F(n + 1) - 49/6 F(n) F(n + 1) - 19/3 F(n) F(n + 1) 2153 6 2 3 - ---- F(n) F(n + 1) + 59/2 F(n) F(n + 1) + 1/4 F(n) - 299/4 F(n + 1) 12 3 1961 3 4 + 3/4 F(n) + ---- F(n) F(n + 1) 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1037 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 247 681 4 31 5 53149 4 62 5 G(n) = --- F(n + 1) + ---- F(n + 1) - --- F(n + 1) - ----- F(n) - --- F(n) 638 2552 319 5104 319 15365 6 2 86315 7 7669 5 3 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 464 2552 2552 4947 3 5 699 8 11519 8 + ---- F(n) F(n + 1) - ---- F(n + 1) + ----- F(n) 1276 5104 1276 155 2 3 155 4 2143 + --- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- 319 319 5104 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1038 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 7 4 3 G(n) = 1/60 F(n) (-15 F(n) - 64 F(n + 1) + 24 F(n + 1) + 1450 F(n) F(n + 1) 5 2 6 2 - 1637 F(n) F(n + 1) + 870 F(n) F(n + 1) + 12 F(n) F(n + 1) 3 4 3 - 565 F(n) F(n + 1) + 30 F(n + 1) - 105 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1039 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 185 9 27 185 2 309 2 411 5 G(n) = ---- F(n + 1) + -- F(n) + --- F(n + 1) + --- F(n) + --- F(n) 638 22 638 638 319 12193 4 5 18363 8 15467 7 2 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 638 638 319 474 6 3 12688 5 4 445 8 - --- F(n) F(n + 1) + ----- F(n) F(n + 1) + --- F(n) F(n + 1) 11 319 319 124 409 4 - --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1040 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 9 3 G(n) = 8855/6 F(n + 1) + 5/2 F(n) - 8855/6 F(n + 1) + 1/2 F(n) 3 2 2 8 + 18359/6 F(n) F(n + 1) - 3/4 F(n) F(n + 1) - 14605/4 F(n) F(n + 1) 3 6 2 4 + 17665/6 F(n) F(n + 1) + 3/4 F(n) F(n + 1) - 4511/2 F(n) F(n + 1) 2 3 2695 2 7 17345 4 - 7081/4 F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 12 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1041 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 871513 7 2 3547 8 758200 5 4 G(n) = ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 18183 18183 18183 4084393 8 709271 4 5 11695 6 3 - ------- F(n) F(n + 1) - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 145464 48488 228 62737 23537 5 25314 5 1070 3339 9 + ------ F(n + 1) - ----- F(n + 1) + ----- F(n) - ---- + ----- F(n + 1) 145464 72732 6061 6061 48488 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1042 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 j) F(3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 5 9 G(n) = 1/616 F(n) (2254900 F(n) F(n + 1) - 3696 F(n) + 425900 F(n + 1) 5 4 - 849923 F(n + 1) + 423907 F(n + 1) - 386447 F(n) F(n + 1) 3 6 3 2 + 821000 F(n) F(n + 1) - 974384 F(n) F(n + 1) 2 7 2 3 - 2483700 F(n) F(n + 1) + 772443 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1043 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 j) F(3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 1989 8 3902 7 61079 5 3 4265 G(n) = ---- F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- 638 957 1914 3828 20 79 16439 4 12395 8 + --- F(n + 1) + --- F(n) - ----- F(n + 1) - ----- F(n) 319 638 3828 3828 47843 6 2 5782 7 7357 2 6 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 1914 957 1276 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1044 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 j) F(3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 55797169 9 454165951 6 4 36874 10 G(n) = -------- F(n) F(n + 1) - --------- F(n) F(n + 1) - ----- F(n) 1242505 2485010 6061 31736 144513179 8 2 - ----- F(n) F(n + 1) - --------- F(n) F(n + 1) 3895 2485010 10211881 9 210299856 7 3 + -------- F(n) F(n + 1) + --------- F(n) F(n + 1) 248501 1242505 1139 5 1070 508 4532362 2 - ---- F(n) F(n + 1) - ---- F(n + 1) + ---- F(n) - ------- F(n + 1) 5945 6061 6061 1242505 60028113 6 50524689 10 + -------- F(n + 1) - -------- F(n + 1) 2485010 2485010 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1045 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 j) F(3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 871330 3 6 70745 2 7 199509 2 3 G(n) = ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 319 319 638 426846 5 11584 2591 39 2 - ------ F(n + 1) - ----- F(n + 1) + ---- F(n) - --- F(n) 319 319 638 638 20 2 438410 9 79 + --- F(n + 1) + ------ F(n + 1) + --- F(n) F(n + 1) 319 319 319 1085890 8 369089 4 57670 3 2 - ------- F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 319 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1046 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 j) F(3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 129 2 5 16 3 4 115 5 2 G(n) = ---- F(n) F(n + 1) + -- F(n) F(n + 1) - --- F(n) F(n + 1) 116 29 638 2 157 3 283 6 237 7 + 7/44 F(n + 1) - ---- F(n + 1) - ---- F(n + 1) + ---- F(n + 1) 1276 1276 1276 59 7 1973 10 18941 4 2 947 4 3 + --- F(n) - ---- F(n) - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 1276 1276 678 3 3 1603 2 4 19373 6 4 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 1276 319 63721 7 3 50502 8 2 86 9 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - -- F(n) F(n + 1) 319 319 29 119 6 25221 5 + --- F(n) F(n + 1) + ----- F(n) F(n + 1) 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1047 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 503145 3120373 11 21207357 11 G(n) = ------- - -------- F(n + 1) + -------- F(n) 2412278 19298224 2412278 541222873 4 3 983515749 9 2 - --------- F(n) F(n + 1) + --------- F(n) F(n + 1) 9649112 1206139 11029882837 8 3 832747901 7 4 - ----------- F(n) F(n + 1) - --------- F(n) F(n + 1) 19298224 4824556 749958213 6 731092215 4 7 - --------- F(n) F(n + 1) + --------- F(n) F(n + 1) 2412278 19298224 587746113 10 2363953 3 4 9226585 3 + --------- F(n) F(n + 1) + ------- F(n) F(n + 1) - -------- F(n + 1) 2412278 438596 19298224 4160899 7 + ------- F(n + 1) 9649112 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1048 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 j) F(4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7060 5 4 2118 4 1049357 5 G(n) = ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + ------- F(n) F(n + 1) 6061 6061 60610 293239 5 1923644 5 5 706 6 3 + ------ F(n) F(n + 1) + ------- F(n) F(n + 1) + ---- F(n) F(n + 1) 6061 30305 6061 214283 6 4 8825 7 2 880070 7 3 - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) 6061 6061 6061 3530 8 2496779 8 2 353 8 + ---- F(n) F(n + 1) - ------- F(n) F(n + 1) + ---- F(n) F(n + 1) 6061 12122 6061 163935 9 4236 2 3 2209 - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n + 1) 6061 6061 12122 45757 2 1412 5 374562 10 109979 6 + ----- F(n + 1) + ---- F(n + 1) + ------ F(n + 1) - ------ F(n + 1) 6061 6061 30305 5510 37425 6 1059 5 - ----- F(n) + ---- F(n) 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1049 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 j) F(4 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1778579 11 2141309 7 167 2504039 3 G(n) = -------- F(n + 1) + ------- F(n + 1) + --- F(n) - ------- F(n + 1) 7140 3570 924 7140 8149 11 62247 2 9 737159 2 + ---- F(n) - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 924 68 7140 8964479 6 1354753 10 458937 2 - ------- F(n) F(n + 1) + ------- F(n) F(n + 1) - ------ F(n) F(n + 1) 7140 1428 2380 447259 2 5 1233191 9 2 + ------ F(n) F(n + 1) + ------- F(n) F(n + 1) 1428 1428 899791 10 + ------ F(n) F(n + 1) 7140 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1050 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 j) F(4 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 321892 2 43067891 6 G(n) = ------ F(n) F(n + 1) + -------- F(n) F(n + 1) 551 6061 67354975 10 1239135 2 - -------- F(n) F(n + 1) - ------- F(n) F(n + 1) 1102 12122 2296652 2 5 25435725 3 8 615 2 - ------- F(n) F(n + 1) + -------- F(n) F(n + 1) + ----- F(n + 1) 319 551 12122 266163503 7 1503 2 14419756 3 - --------- F(n + 1) - ----- F(n) - -------- F(n + 1) 12122 12122 6061 2118 13409200 11 110601 3 + ---- F(n) F(n + 1) + -------- F(n + 1) + ------ F(n) 6061 551 12122 7330925 2 9 95859771 3 4 + ------- F(n) F(n + 1) + -------- F(n) F(n + 1) 1102 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1051 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 631 6 G(n) = -1/22 F(n) + 7/11 F(n) F(n + 1) - --- F(n) F(n + 1) 22 183 2 717 2 5 3 3 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - 5/11 F(n) F(n + 1) 22 11 243 3 4 1503 4 3 302 2 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 22 22 11 23 3 7 5 2 4 + -- F(n) + 5/22 F(n + 1) + 5/11 F(n) F(n + 1) + 5/22 F(n) F(n + 1) 22 4 2 6 - 5/22 F(n) F(n + 1) - 5/22 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1052 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 j) F(n + j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 2 5 G(n) = 1/12 F(n) (685 F(n) F(n + 1) - 1520 F(n) F(n + 1) 4 3 2 + 1810 F(n) F(n + 1) + 2 F(n + 1) - 679 F(n) F(n + 1) 2 3 4 3 3 + 180 F(n) F(n + 1) - F(n) - 450 F(n) F(n + 1) - 23 F(n) - 4 F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1053 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 j) F(n + j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 2 20 9 135 2 89316 2 3 G(n) = --- F(n + 1) - --- F(n + 1) + --- F(n) - ----- F(n) F(n + 1) 319 319 638 319 209030 8 438430 4 5 5530 3 6 - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 68047 3 2 509175 2 7 1779 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + ---- F(n) 638 319 638 10047 4 103 7181 4 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 638 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1054 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 j) F(n + j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 26 280 2 6 13875 3 5 8295 7 G(n) = - -- + --- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 11 22 11 21 2 3 3 3354 4 + -- F(n) F(n + 1) + 4/11 F(n) - 5/22 F(n + 1) - ---- F(n + 1) 22 11 614 2 2 2 1466 3 - --- F(n) F(n + 1) - 7/11 F(n) F(n + 1) + ---- F(n) F(n + 1) 11 11 218 3 8 + --- F(n) F(n + 1) + 615/2 F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1055 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 j) F(n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 8 G(n) = -1/12 F(n) (17710 F(n) F(n + 1) + 8395 F(n + 1) + F(n) F(n + 1) 3 2 2 3 + 3545 F(n) F(n + 1) - 1318 F(n) F(n + 1) + 379 F(n) F(n + 1) 4 2 7 - 8360 F(n + 1) - 32 - 4 F(n) - 20405 F(n) F(n + 1) 2 6 2 + 90 F(n) F(n + 1) - F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1056 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 j) F(n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 19 2 7 2 6 189 4 5 G(n) = -- F(n) F(n + 1) - 2 F(n) F(n + 1) - --- F(n) F(n + 1) 22 22 108 4 4 108 6 2 21 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + 5/22 F(n + 1) - -- F(n) 11 11 22 141 3 38 3 2 327 5 4 + --- F(n) F(n + 1) - -- F(n) F(n + 1) + --- F(n) F(n + 1) 22 11 11 604 6 3 48 7 1069 7 2 - --- F(n) F(n + 1) - -- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 11 22 367 8 3 5 3 - --- F(n) F(n + 1) + 9/11 F(n) F(n + 1) - 5/22 + 4/11 F(n) F(n + 1) 22 8 48 9 - 2/11 F(n) + -- F(n) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1057 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 185 2 199857 2 3 1281 22559 G(n) = --- F(n + 1) - ------ F(n) F(n + 1) + ---- F(n) - ----- F(n + 1) 638 638 319 638 70745 2 7 871330 3 6 57119 3 2 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 319 2 1085890 8 124 - 5/319 F(n) - ------- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 370945 4 438410 9 427223 5 + ------ F(n) F(n + 1) + ------ F(n + 1) - ------ F(n + 1) 638 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1058 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 j) F(n + 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 3 G(n) = -1/12 F(n) (-8340 F(n + 1) + 3505 F(n) F(n + 1) + 427 F(n) F(n + 1) 2 3 5 8 - 5 F(n + 1) + 17710 F(n) F(n + 1) + 8395 F(n + 1) + 5 F(n) F(n + 1) 7 2 2 2 6 - 20405 F(n) F(n + 1) - 1334 F(n) F(n + 1) + 90 F(n) F(n + 1) - 48) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1059 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 j) F(n + 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 114 4 3 1033 2 1673 5 2 G(n) = --- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 1276 1276 63923 4 6 289 6 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - 1/1276 F(n) F(n + 1) 2552 1276 1959 5 8731 3 3 1673 3 4 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 1276 1276 1276 47007 9 156369 8 2 61 6 + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) + --- F(n) F(n + 1) 1276 2552 638 421131 6 4 85203 4 2 185 3 - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + --- F(n + 1) 2552 2552 638 57 7 1857 10 185 10 2151 7 3 - --- F(n) - ---- F(n) - --- F(n + 1) + ---- F(n) F(n + 1) 319 319 638 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1060 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 j) F(n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 G(n) = 1/2976 F(n) (617754 F(n) F(n + 1) - 2976 F(n) F(n + 1) 9 5 3 + 490328 F(n + 1) - 502899 F(n + 1) + 1984 F(n + 1) 2 7 8 2 3 + 942297 F(n) F(n + 1) + 768013 F(n) F(n + 1) - 903318 F(n) F(n + 1) 8 3 2 - 1404906 F(n) F(n + 1) - 167122 F(n) - 248 F(n) + 1488 F(n) F(n + 1) 9 + 149514 F(n) + 10091 F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1061 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 106193 3 2905 2 2296652 2 5 G(n) = ------ F(n) + ----- F(n) - ------- F(n) F(n + 1) 12122 12122 319 266163503 7 1070 2 28836757 3 - --------- F(n + 1) - ---- F(n + 1) - -------- F(n + 1) 12122 6061 12122 43067891 6 7082175 2 1016 + -------- F(n) F(n + 1) + ------- F(n) F(n + 1) + ---- F(n) F(n + 1) 6061 12122 6061 25435725 3 8 618741 2 + -------- F(n) F(n + 1) - ------ F(n) F(n + 1) 551 6061 95859771 3 4 7330925 2 9 + -------- F(n) F(n + 1) + ------- F(n) F(n + 1) 12122 1102 67354975 10 13409200 11 - -------- F(n) F(n + 1) + -------- F(n + 1) 1102 551 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1062 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 j) F(n + 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 44637 9 25469 3 7 756911 5 G(n) = ------ F(n) F(n + 1) - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 2552 1276 6380 496537 153876 5 21 6 + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) - -- F(n) F(n + 1) 12760 1595 29 21 5 2 30013 2 191 3 317 3 + -- F(n) F(n + 1) - ----- F(n + 1) - --- F(n + 1) - --- F(n) 29 1595 638 638 3511 6 21 7 29913 10 328457 9 - ---- F(n) + -- F(n + 1) + ----- F(n + 1) + ------ F(n) F(n + 1) 638 58 1595 2552 4146723 5 5 149521 4 6 21 6 + ------- F(n) F(n + 1) - ------ F(n) F(n + 1) + -- F(n) F(n + 1) 12760 638 58 12 2 - --- F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1063 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 j) F(n + 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 63609 7 959 4 53029 3 101 7 G(n) = - 5/116 + ----- F(n) + ---- F(n) - ----- F(n) - ---- F(n + 1) 1276 1276 1276 1276 2605 3 5 421037 5 2 2605 5 3 - ---- F(n) F(n + 1) + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 2552 1276 2552 2 9 2591 3 49769 3 4 - 5/44 F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 2552 232 7661745 7 4 7037125 5 6 21 3 + ------- F(n) F(n + 1) - ------- F(n) F(n + 1) + ---- F(n + 1) 2552 2552 1276 135 4 521 2 6 521 7 + ---- F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 1276 1276 2552 1534639 4 3 667925 3 8 - ------- F(n) F(n + 1) + ------ F(n) F(n + 1) 1276 2552 182675 4 7 + ------ F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1064 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 9 54 15 4 845 4 G(n) = 5/22 F(n + 1) + -- F(n) - -- F(n) - 5/22 - --- F(n) F(n + 1) 11 22 22 6933 4 5899 2 3 2 2 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - 3/2 F(n) F(n + 1) 11 22 14015 8 3 14605 4 5 - ----- F(n) F(n + 1) + 9/11 F(n) F(n + 1) - ----- F(n) F(n + 1) 22 11 3 6 1184 3 2 21 3 - 25 F(n) F(n + 1) + ---- F(n) F(n + 1) + -- F(n) F(n + 1) 11 11 34165 2 7 + ----- F(n) F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1065 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 j) F(2 n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 3931 2 3 G(n) = 1/4 F(n + 1) - 23/4 F(n) - ---- F(n + 1) + 7/6 F(n) F(n + 1) 12 9 5 2 4 - 46700/3 F(n) F(n + 1) + 2372 F(n) F(n + 1) - 6537/4 F(n) F(n + 1) 3 3 3 7 2 2 + 2595/2 F(n) F(n + 1) + 36325/3 F(n) F(n + 1) - 3/4 F(n) F(n + 1) 74875 10 6 + 359/6 F(n) F(n + 1) - 1/4 + ----- F(n + 1) - 5912 F(n + 1) 12 3 2 8 - 1/6 F(n) F(n + 1) + 5475/4 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1066 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 j) F(2 n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 25 5 364025 2 8 3 7 G(n) = -- F(n + 1) + ------ F(n) F(n + 1) - 3175 F(n) F(n + 1) 44 44 2 61261 6 80 6 14 5 - 5567/4 F(n + 1) + ----- F(n + 1) - -- F(n) + -- F(n) 22 11 11 61275 10 63 3 2 36700 3 3 - ----- F(n + 1) + -- F(n) F(n + 1) + ----- F(n) F(n + 1) 44 11 11 4 52617 4 2 306975 4 6 - 11/4 F(n) F(n + 1) + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 44 44 1125 5 2 3 118741 2 4 + ---- F(n) F(n + 1) - 13/4 F(n) F(n + 1) - ------ F(n) F(n + 1) 22 44 35 - -- F(n + 1) 44 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1067 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 64189 2 91 2 2 4534583 2 5 G(n) = ------ F(n) F(n + 1) + -- F(n) F(n + 1) - ------- F(n) F(n + 1) 638 58 638 5001893 3 4 14474300 3 8 + ------- F(n) F(n + 1) + -------- F(n) F(n + 1) 638 319 370005 2 175 3 2217778 6 + ------ F(n) F(n + 1) + --- F(n) F(n + 1) + ------- F(n) F(n + 1) 638 319 319 13764099 7 19179400 10 423 753268 3 - -------- F(n + 1) - -------- F(n) F(n + 1) + --- - ------ F(n + 1) 638 319 638 319 651 3 119 4 5319 3 - --- F(n) F(n + 1) - --- F(n + 1) + ---- F(n) 319 319 638 2098075 2 9 7635225 11 + ------- F(n) F(n + 1) + ------- F(n + 1) 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1068 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 j) F(2 n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 59 3 4 2 9 365 6 3 G(n) = --- F(n) F(n + 1) + 21/8 F(n) F(n + 1) + --- F(n) F(n + 1) 12 96 4163 6 5 6 115 5 4 + ---- F(n) F(n + 1) + 2122/3 F(n) F(n + 1) - --- F(n) F(n + 1) 20 16 7853 4 7 4 5 7003 4 3 - ---- F(n) F(n + 1) - 45/8 F(n) F(n + 1) + ---- F(n) F(n + 1) 24 24 25 4 155 3 6 45 7 2 - -- F(n) F(n + 1) + --- F(n) F(n + 1) + -- F(n) F(n + 1) 96 24 16 21973 6 5 10 49 5 - ----- F(n) F(n + 1) - 2111/3 F(n) F(n + 1) + -- F(n + 1) 120 48 3 35 9 21 3217 7 5 + 3/8 F(n + 1) - -- F(n + 1) - -- F(n + 1) + ---- F(n) + 7/12 F(n) 96 32 15 7 4121 11 - 3/8 F(n + 1) - ---- F(n) 20 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1069 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 253260 7 10 235 3 37 2 G(n) = ------- F(n + 1) - 5/22 F(n + 1) + --- F(n) - -- F(n) 11 22 22 2 4 100 5 70775 2 9 + 20 F(n) F(n + 1) - --- F(n) F(n + 1) + ----- F(n) F(n + 1) 11 11 175 3 3 98969 2 5 2 8 + --- F(n) F(n + 1) - ----- F(n) F(n + 1) + 5/22 F(n) F(n + 1) 11 22 1218 2 636550 10 52487 4 3 - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 11 11 22 9 755 4 2 958325 3 8 + 5/11 F(n) F(n + 1) - --- F(n) F(n + 1) + ------ F(n) F(n + 1) 22 22 4 6 66837 3 4 - 5/22 F(n) F(n + 1) + ----- F(n) F(n + 1) + 11 F(n) F(n + 1) 22 6617 2 3 7 506525 11 + ---- F(n) F(n + 1) - 5/11 F(n) F(n + 1) + ------ F(n + 1) 11 22 246771 6 + ------ F(n) F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1070 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 7 5 G(n) = -1/77 F(n) (616 F(n) + 28600 F(n) F(n + 1) - 223389 F(n + 1) 3 2 9 + 26630 F(n) F(n + 1) - 5260 F(n + 1) + 228625 F(n + 1) 3 6 4 8 + 461100 F(n) F(n + 1) + 95399 F(n) F(n + 1) - 563725 F(n) F(n + 1) 2 3 - 48596 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1071 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 25271800 2 9 674615181 3 4 G(n) = -------- F(n) F(n + 1) + --------- F(n) F(n + 1) 3781 83182 1832238513 7 102733786 3 185245675 11 - ---------- F(n + 1) - --------- F(n + 1) + --------- F(n + 1) 83182 41591 7562 497262 3 1830 232609725 10 + ------ F(n) + ----- - --------- F(n) F(n + 1) 41591 41591 3781 4832093 2 300857313 2 5 - ------- F(n) F(n + 1) - --------- F(n) F(n + 1) 41591 41591 351466025 3 8 51847985 2 + --------- F(n) F(n + 1) + -------- F(n) F(n + 1) 7562 83182 579229197 6 + --------- F(n) F(n + 1) 83182 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1072 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1070 10 48445 2 416577 G(n) = ---- F(n + 1) - ----- F(n) + ------ F(n) F(n + 1) 6061 6061 6061 17291905 5 2140 8 17713790 9 + -------- F(n) F(n + 1) + ---- F(n) F(n + 1) - -------- F(n) F(n + 1) 6061 6061 6061 1070 2 3 7376396 2 4 1070 2 7 + ---- F(n) F(n + 1) - ------- F(n) F(n + 1) + ---- F(n) F(n + 1) 6061 6061 6061 87370585 2 8 2140 3 2 + -------- F(n) F(n + 1) - ---- F(n) F(n + 1) 12122 6061 3780336 3 3 2259735 3 7 1070 4 + ------- F(n) F(n + 1) - ------- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 6061 6061 4184993 4 2 1070 4 5 - ------- F(n) F(n + 1) - ---- F(n) F(n + 1) 12122 6061 35683205 4 6 43 2140 4 - -------- F(n) F(n + 1) - ---- F(n) + ---- F(n) F(n + 1) 6061 6061 6061 2140 3 6 1070 9 - ---- F(n) F(n + 1) - ---- F(n + 1) 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1073 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 272425 11 190979 3 2788 3 G(n) = ------ F(n + 1) - ------ F(n + 1) + ---- F(n) 11 77 231 518825 2 9 1884506 3 4 16 + ------ F(n) F(n + 1) + ------- F(n) F(n + 1) - --- F(n) 77 231 231 233446 6 4788650 10 26816 2 + ------ F(n) F(n + 1) - ------- F(n) F(n + 1) - ----- F(n) F(n + 1) 33 77 231 1687201 2 5 3619300 3 8 48064 2 - ------- F(n) F(n + 1) + ------- F(n) F(n + 1) + ----- F(n) F(n + 1) 231 77 77 1715996 7 - ------- F(n + 1) 77 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1074 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 706350 2 4454614 3 4 G(n) = ------- F(n) F(n + 1) + ------- F(n) F(n + 1) 6061 551 25435725 3 8 1070 2 13409200 11 + -------- F(n) F(n + 1) - ---- F(n + 1) + -------- F(n + 1) 551 6061 551 73759 3 1027 2 417117 2 5 + ----- F(n) - ---- F(n) - ------ F(n) F(n + 1) 6061 6061 58 7330925 2 9 29885861 3 67354975 10 + ------- F(n) F(n + 1) - -------- F(n + 1) - -------- F(n) F(n + 1) 1102 12122 1102 86 3779946 2 7596551 6 - ---- F(n) F(n + 1) + ------- F(n) F(n + 1) + ------- F(n) F(n + 1) 6061 6061 1102 24101309 7 - -------- F(n + 1) 1102 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1075 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 10 G(n) = -1/40 F(n) (-7788 F(n) F(n + 1) + 720 F(n) - 5263425 F(n) F(n + 1) 2 3 4 2 9 + 44618 F(n) F(n + 1) + 624859 F(n) F(n + 1) + 535650 F(n) F(n + 1) 3 8 7 11 + 4015475 F(n) F(n + 1) - 1915842 F(n + 1) + 2100450 F(n + 1) 3 2 5 6 - 184596 F(n + 1) - 598794 F(n) F(n + 1) + 648673 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1076 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 j) F(4 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 7705447 3 3 12585 3 2 8098200 9 G(n) = ------- F(n) F(n + 1) - ----- F(n) F(n + 1) - ------- F(n) F(n + 1) 6061 12122 551 6282425 3 7 9455469 2 4 + ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 551 6061 26776509 5 12585 4 1975276 2 + -------- F(n) F(n + 1) + ----- F(n) F(n + 1) - ------- F(n + 1) 12122 12122 6061 76575 2 8 63 2517 5 2517 5 + ----- F(n) F(n + 1) + --- F(n + 1) - ---- F(n + 1) + ----- F(n) 58 209 6061 12122 75249 2 369571 3244025 10 - ----- F(n) + ------ F(n) F(n + 1) + ------- F(n + 1) 12122 6061 551 33708309 6 - -------- F(n + 1) 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1077 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 j) F(4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 11557 7 30179 10 99201 2 G(n) = ------ F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 300 60 700 8896 2 287768 6 42061 9 2 + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 75 525 84 193343 8 3 440353 10 573763 11 - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) - ------ F(n + 1) 140 2100 2100 11075 11 327331 3 13 322559 2 5 + ----- F(n) + ------ F(n + 1) + --- F(n) + ------ F(n) F(n + 1) 924 1050 924 420 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1078 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 j) F(4 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2994 436485765 4 8 1830 4 5 G(n) = ------ F(n) - --------- F(n) F(n + 1) + ----- F(n) F(n + 1) 41591 4378 41591 1830 4 647376175 4 4 + ----- F(n) F(n + 1) - --------- F(n) F(n + 1) 41591 41591 554439410 3 9 3660 3 6 - --------- F(n) F(n + 1) + ----- F(n) F(n + 1) 41591 41591 619068050 3 5 3660 3 2 + --------- F(n) F(n + 1) + ----- F(n) F(n + 1) 41591 41591 5436918205 2 10 8108746 3 + ---------- F(n) F(n + 1) + ------- F(n) F(n + 1) 41591 41591 1337129715 2 6 1830 2 7 - ---------- F(n) F(n + 1) - ----- F(n) F(n + 1) 83182 41591 1830 2 3 4339073305 11 - ----- F(n) F(n + 1) - ---------- F(n) F(n + 1) 41591 83182 46359420 2 2 3956192035 7 - -------- F(n) F(n + 1) + ---------- F(n) F(n + 1) 41591 83182 3660 8 3660 4 10076258 3 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + -------- F(n) F(n + 1) 41591 41591 2189 745644 4 1830 9 1830 12 - ------ F(n) + ----- F(n + 1) - ----- F(n + 1) 41591 41591 41591 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1079 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 j) F(4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 8905903 3 4 704697 2 G(n) = ------- F(n) F(n + 1) - ------ F(n) F(n + 1) 1102 6061 67354975 10 3800983 6 - -------- F(n) F(n + 1) + ------- F(n) F(n + 1) 1102 551 7541975 2 2517 24103399 7 + ------- F(n) F(n + 1) + ---- F(n) F(n + 1) - -------- F(n + 1) 12122 6061 1102 29863631 3 208591 2 5 13409200 11 - -------- F(n + 1) - ------ F(n) F(n + 1) + -------- F(n + 1) 12122 29 551 149361 3 3897 2 690 2 7330925 2 9 + ------ F(n) - ----- F(n) - ---- F(n + 1) + ------- F(n) F(n + 1) 12122 12122 6061 1102 25435725 3 8 + -------- F(n) F(n + 1) 551 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1080 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 j) F(4 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 5 2 G(n) = -1/924 F(n) (-13881880 F(n) F(n + 1) - 179906 F(n) F(n + 1) 10 6 - 122088000 F(n) F(n + 1) + 15065700 F(n) F(n + 1) 3 4 2 9 + 14468560 F(n) F(n + 1) + 12409500 F(n) F(n + 1) 2 11 7 + 1031720 F(n) F(n + 1) + 48723000 F(n + 1) - 44450860 F(n + 1) 3 3 8 3 - 4271940 F(n + 1) - 26 F(n + 1) + 93157500 F(n) F(n + 1) + 16619 F(n) + 13 F(n)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1081 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 j) F(4 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3207 3 690 12 112305 4 690 11 G(n) = ---- F(n) + ---- F(n + 1) - ------ F(n) - ---- F(n + 1) 6061 6061 6061 6061 286271695 7 193134375 2 6 + --------- F(n) F(n + 1) - --------- F(n) F(n + 1) 6061 12122 690 2 5 787225310 2 10 1380 3 4 + ---- F(n) F(n + 1) + --------- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 6061 6061 9443845 3 5 1380 3 8 1380 10 + ------- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 6061 6061 11691 2 6748462 2 2 - ----- F(n) F(n + 1) - ------- F(n) F(n + 1) 12122 6061 188063475 4 4 314113255 11 - --------- F(n) F(n + 1) - --------- F(n) F(n + 1) 12122 6061 1380 6 690 2 9 2372009 3 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ------- F(n) F(n + 1) 6061 6061 12122 7551 2 55668781 3 + ----- F(n) F(n + 1) + -------- F(n) F(n + 1) 12122 12122 80447120 3 9 690 4 3 690 4 7 - -------- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 6061 6061 1200429995 4 8 - ---------- F(n) F(n + 1) 12122 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1082 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 7 7 G(n) = -1/24 F(n) (-361 F(n) + 2 F(n) - 450 F(n + 1) + 407 F(n) 3 6 2 5 + 442 F(n + 1) + 1565 F(n) F(n + 1) - 4 F(n + 1) - 1394 F(n) F(n + 1) 2 2 - 653 F(n) F(n + 1) + 446 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1083 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 j) F(n + j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2663 9 17657 8049 5886 5 G(n) = ----- F(n + 1) - ----- F(n) + ---- F(n + 1) - ---- F(n + 1) 7975 7975 7975 7975 91671 8 343993 7 2 623779 6 3 - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 7975 7975 7975 300051 5 4 92 2 51857 9 3346 4 + ------ F(n) F(n + 1) - --- F(n) + ----- F(n) + ---- F(n) F(n + 1) 7975 319 7975 725 108 20 2 + --- F(n) F(n + 1) + --- F(n + 1) 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1084 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 j) F(n + j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3529 5 2 2 5 G(n) = 23/6 F(n) + ---- F(n + 1) - 1/4 F(n) F(n + 1) - 7/8 F(n) F(n + 1) 12 6 4 5 49205 2 7 - 1/4 F(n) F(n + 1) - 14605/8 F(n) F(n + 1) + ----- F(n) F(n + 1) 24 7 3 1337 5 4 + 1/8 F(n + 1) + 1/6 F(n) + ---- F(n + 1) + 7705/6 F(n) F(n + 1) 24 8395 9 3 4 3 - ---- F(n + 1) + 1/2 F(n) F(n + 1) - 1/8 F(n + 1) 24 3 2 8485 3 6 4 - 1426/3 F(n) F(n + 1) - ---- F(n) F(n + 1) - 991/3 F(n) F(n + 1) 12 4 3 + 7/8 F(n) F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1085 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 j) F(n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 3 119950 3 7 208317 4 2 G(n) = --- F(n + 1) - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 319 319 638 1848250 4 6 39519 898506 5 - ------- F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 319 638 319 918425 9 465 2 773545 2 4 - ------ F(n) F(n + 1) - --- F(n) F(n + 1) - ------ F(n) F(n + 1) 319 638 638 353 2 390171 3 3 2266175 2 8 + --- F(n) F(n + 1) + ------ F(n) F(n + 1) + ------- F(n) F(n + 1) 638 638 319 263 3 20 6 4091 2 + --- F(n) - --- F(n + 1) - ---- F(n) 638 319 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1086 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 j) F(n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 3 6 G(n) = 1/36 F(n) (-41 F(n) F(n + 1) - 11511 F(n) F(n + 1) 8 9 2 + 8594 F(n) F(n + 1) + 6 F(n + 1) + 24 F(n) F(n + 1) 2 3 8 4 - 9072 F(n) F(n + 1) - 2392 F(n) F(n + 1) + 4170 F(n) F(n + 1) 2 2 7 3 - 18 F(n) F(n + 1) + 10444 F(n) F(n + 1) - 1805 F(n) - 9 F(n) 3 9 + 12 F(n + 1) + 1598 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1087 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 j) F(n + 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2012819 3 4 401 2 145865 3 G(n) = ------- F(n) F(n + 1) - ----- F(n) + ------ F(n) 638 12122 12122 15207285 3 8 2140 3 7 2140 3 3 - -------- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 6061 6061 375640435 2 9 1070 2 4 1070 2 8 + --------- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 12122 6061 6061 147500130 4 7 1070 4 6 - --------- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 6061 2713891 4 3 1070 4 2 7539505 2 - ------- F(n) F(n + 1) - ---- F(n) F(n + 1) + ------- F(n) F(n + 1) 1102 6061 12122 465 703595 2 150904205 10 + ---- F(n) F(n + 1) - ------ F(n) F(n + 1) - --------- F(n) F(n + 1) 6061 6061 12122 2140 9 71676534 6 2140 5 + ---- F(n) F(n + 1) + -------- F(n) F(n + 1) + ---- F(n) F(n + 1) 6061 6061 6061 57343937 2 5 1070 10 1070 11 - -------- F(n) F(n + 1) - ---- F(n + 1) + ---- F(n + 1) 12122 6061 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1088 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 j) F(n + 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 1848250 10 5425 2 3475869 6 G(n) = ------- F(n + 1) - ---- F(n) - ------- F(n + 1) 319 638 638 417925 2 8 491888 2 4 37 2 + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) - -- F(n) F(n + 1) 319 319 58 4614925 9 675457 5 44333 - ------- F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 638 3576550 3 7 834645 3 3 321 3 + ------- F(n) F(n + 1) + ------ F(n) F(n + 1) + --- F(n) 319 638 638 295 2 20061 2 20 3 + --- F(n) F(n + 1) - ----- F(n + 1) + --- F(n + 1) 638 58 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1089 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 j) F(n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 11 20 545 4 134790 10 G(n) = ---- F(n + 1) + --- - --- F(n) - ------ F(n) F(n + 1) 319 319 638 11 75731 2 629 2 2 1478368 2 5 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) - ------- F(n) F(n + 1) 638 638 319 9733320 2 9 91 3 993439 3 4 + ------- F(n) F(n + 1) + -- F(n) F(n + 1) + ------ F(n) F(n + 1) 319 58 319 1566363 4 3 7635245 4 7 - ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 638 319 398251 2 227 3 7419309 6 + ------ F(n) F(n + 1) + --- F(n) F(n + 1) + ------- F(n) F(n + 1) 638 319 638 8161 3 796190 3 8 + ---- F(n) - ------ F(n) F(n + 1) 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1090 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1788385 8 16 2 1059791 3 G(n) = -------- F(n + 1) + --- F(n) + ------- F(n) F(n + 1) 21 231 231 509750 3 5 14492125 3 9 + ------ F(n) F(n + 1) + -------- F(n) F(n + 1) 11 77 19586500 11 352585 7 53440 3 - -------- F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 77 21 231 2410875 2 10 1054795 2 6 + ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 77 33 87445 2 2 32 4174 172046 4 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- - ------ F(n + 1) 77 231 231 11 7763125 12 + ------- F(n + 1) 77 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1091 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 j) F(n + 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3062772001 12 1070 3 8025451399 3 G(n) = ----------- F(n + 1) - ---- F(n + 1) + ---------- F(n) F(n + 1) 2733511 6061 5467022 422 2 1586125069 11 1428 2 + ---- F(n) F(n + 1) + ---------- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 497002 6061 4262322660 7 2329148061 2 2 - ---------- F(n) F(n + 1) + ---------- F(n) F(n + 1) 2733511 5467022 36158986713 2 6 2929701135 12 1417 3 + ----------- F(n) F(n + 1) - ---------- F(n) + ----- F(n) 10934044 10934044 12122 23456214585 2 10 7524055621 11 - ----------- F(n) F(n + 1) - ---------- F(n) F(n + 1) 10934044 5467022 3025371169 10 2 10284257693 8 2731610209 - ---------- F(n) F(n + 1) + ----------- F(n + 1) + ---------- 994004 10934044 10934044 381424809 4 - --------- F(n + 1) 5467022 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1092 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 9 7 2 5 G(n) = -1/24 F(n) (-5 F(n + 1) + 186 F(n) + 6 F(n) - 46 F(n) F(n + 1) 5 3 4 2 7 + F(n + 1) + 38 F(n) F(n + 1) - 149 F(n) F(n + 1) 4 3 3 6 4 5 + 34 F(n) F(n + 1) + 838 F(n) F(n + 1) - 2198 F(n) F(n + 1) 7 2 6 3 5 4 + 2270 F(n) F(n + 1) - 3474 F(n) F(n + 1) + 3432 F(n) F(n + 1) 5 2 7 3 8 - 26 F(n) F(n + 1) + 6 F(n + 1) - 14 F(n + 1) - 895 F(n) F(n + 1) 6 - 4 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1093 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 j) F(2 n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 436 8 256 4 18940 7 14932 3 288 3 G(n) = ---- F(n) + --- F(n) + ----- F(n) - ----- F(n) - --- F(n) F(n + 1) 319 319 319 319 319 1744 6 2 104759 2 195639 3 4 + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 319 319 319 594414 8 3 850611 6 5 873995 2 9 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 319 319 319 2180 2 6 416806 2 5 35698 3 8 - ---- F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 29 872 3 5 333968 10 872 7 + --- F(n) F(n + 1) + ------ F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 39906 6 20 3 656 3 - ----- F(n) F(n + 1) - --- F(n + 1) - --- F(n) F(n + 1) 29 319 319 20 4 + --- F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1094 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 j) F(2 n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 9 2 G(n) = -1/12 F(n) (-139 F(n) - 393550 F(n) F(n + 1) - 5 - 7614 F(n + 1) 6 10 5 - 150587 F(n + 1) + 158200 F(n + 1) + 60528 F(n) F(n + 1) 2 2 2 4 + 1397 F(n) F(n + 1) - 25 F(n) F(n + 1) - 39699 F(n) F(n + 1) 2 8 3 3 3 7 + 31025 F(n) F(n + 1) + 30469 F(n) F(n + 1) + 309975 F(n) F(n + 1) 3 + 25 F(n) F(n + 1)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1095 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(3 j) F(2 n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 623 12107 4 2823195 4 8 G(n) = --- F(n) - ----- F(n) - ------- F(n) F(n + 1) 638 638 29 4190915 3 9 4956675 2 6 1728 3 2 - ------- F(n) F(n + 1) - ------- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 319 63610 3 40766270 2 10 40 3 6 + ----- F(n) F(n + 1) + -------- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 4655060 3 5 20 4 5 4918660 4 4 + ------- F(n) F(n + 1) + --- F(n) F(n + 1) - ------- F(n) F(n + 1) 319 319 319 1668 4 715145 2 2 20 9 - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) + --- F(n + 1) 319 638 319 16262960 11 2924473 3 - -------- F(n) F(n + 1) + ------- F(n) F(n + 1) 319 638 14800525 7 40 8 36 2 3 + -------- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 319 56 4 20 12 20 2 7 - --- F(n) F(n + 1) - --- F(n + 1) - --- F(n) F(n + 1) 319 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1096 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 9 2 3 G(n) = -1/12 F(n) (166250 F(n) F(n + 1) - 70 F(n) F(n + 1) 2 5 2 4 - 187030 F(n) F(n + 1) - 2340 F(n) F(n + 1) - 85 F(n) F(n + 1) 6 10 2 + 206005 F(n) F(n + 1) - 1652625 F(n) F(n + 1) + 13620 F(n) F(n + 1) 5 7 11 + 40 F(n + 1) - 603000 F(n + 1) + 659750 F(n + 1) 3 8 3 + 1262875 F(n) F(n + 1) - 46 F(n + 1) - 56750 F(n + 1) 3 2 3 4 3 + 145 F(n) F(n + 1) + 193045 F(n) F(n + 1) + 208 F(n) + 8 F(n)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1097 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 6889493 7 7857 2 G(n) = -------- F(n + 1) + 9/44 F(n) - ---- F(n) F(n + 1) 308 77 4788650 10 560243 6 180849 2 - ------- F(n) F(n + 1) + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 77 77 308 3619300 3 8 2459321 3 4 738407 3 + ------- F(n) F(n + 1) + ------- F(n) F(n + 1) - ------ F(n + 1) 77 308 308 387 3 1125193 2 5 518825 2 9 + --- F(n) - ------- F(n) F(n + 1) + ------ F(n) F(n + 1) 44 154 77 272425 11 + ------ F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1098 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 21464874 8 3522015 5 17835331 11 G(n) = --------- F(n) - ------- F(n) + -------- F(n) F(n + 1) 1206139 9649112 9649112 27039489395 4 8 735898081 5 3 + ----------- F(n) F(n + 1) + --------- F(n) F(n + 1) 4824556 332728 3522015 5 4 4528305 4 + ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 2412278 4824556 1118637 7 2515725 8 6625020 7 - ------- F(n) F(n + 1) - ------- F(n) F(n + 1) + ------- F(n) F(n + 1) 438596 9649112 41591 16079139895 5 7 1704829451 6 2 - ----------- F(n) F(n + 1) - ---------- F(n) F(n + 1) 1206139 2412278 12578625 6 3 2515725 4 5 + -------- F(n) F(n + 1) - ------- F(n) F(n + 1) 4824556 1206139 2515725 7 2 1175108441 6 6 - ------- F(n) F(n + 1) - ---------- F(n) F(n + 1) 9649112 1206139 122635709355 7 5 2515725 8 1556991 + ------------ F(n) F(n + 1) - ------- F(n) F(n + 1) + ------- F(n) 9649112 4824556 9649112 503145 12 18871079 3 - ------- F(n + 1) + -------- F(n) F(n + 1) 2412278 9649112 27288922511 4 4 503145 9 - ----------- F(n) F(n + 1) + ------- F(n + 1) 4824556 2412278 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1099 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 1417492 13065889 12 5207611 2 2 G(n) = - ------- + -------- F(n + 1) - ------- F(n) F(n + 1) 721 2884 721 8561138 7 7023129 3 - ------- F(n) F(n + 1) + 9/22 F(n) F(n + 1) - ------- F(n) F(n + 1) 721 721 30359061 2 6 17694561 10 2 - -------- F(n) F(n + 1) + -------- F(n) F(n + 1) 1442 721 14175010 9 3 64863598 11 + -------- F(n) F(n + 1) - -------- F(n) F(n + 1) 721 7931 13504416 11 2 5579551 8 + -------- F(n) F(n + 1) - 9/44 F(n) + ------- F(n + 1) 1133 1442 61805105 12 18555023 4 + -------- F(n) - -------- F(n + 1) 31724 2884 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1100 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 105 6 105 9 1953 9 G(n) = -3/11 F(n) F(n + 1) - --- F(n + 1) + --- F(n + 1) + ---- F(n) 638 638 638 39 6 12369 8 15578 7 2 - --- F(n) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 638 638 319 119 3 6 354 2 7 23495 5 4 + --- F(n) F(n + 1) + --- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 638 6811 4 5 421 8 2 4 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) - 9/319 F(n) F(n + 1) 638 638 4 2 126 5 18976 6 3 + 9/319 F(n) F(n + 1) + --- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 18 5 - --- F(n) F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1101 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 j) F(n + j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 13409200 11 243525 2 690 2 G(n) = -------- F(n + 1) + ------ F(n) F(n + 1) - ---- F(n + 1) 551 418 6061 6231 147500510 3 266223733 4 3 + ----- F(n) F(n + 1) - --------- F(n + 1) + --------- F(n) F(n + 1) 12122 6061 12122 314075766 3 4 15385331 6 + --------- F(n) F(n + 1) - -------- F(n) F(n + 1) 6061 418 67354975 10 176731392 2 5 - -------- F(n) F(n + 1) - --------- F(n) F(n + 1) 1102 6061 7330925 2 9 25435725 3 8 110259 3 + ------- F(n) F(n + 1) + -------- F(n) F(n + 1) + ------ F(n) 1102 551 12122 1161 2 618418 2 - ----- F(n) - ------ F(n) F(n + 1) 12122 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1102 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 j) F(n + j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 51 6 30 3 15 2 G(n) = ---- F(n) F(n + 1) + --- F(n) + -- F(n) F(n + 1) 319 319 58 858063 6 4 102 6 60362 5 5 - ------ F(n) F(n + 1) - --- F(n) F(n + 1) + ----- F(n) F(n + 1) 638 319 319 255 5 2 15520 5 75933 4 6 - --- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 58 124901 4 2 16392 3 7 255 3 4 - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + --- F(n) F(n + 1) 638 29 319 359965 3 3 587 2 8 + ------ F(n) F(n + 1) + --- F(n) F(n + 1) - 1/29 F(n) F(n + 1) 638 638 1944 6 105 2 105 3 - ---- F(n) + --- F(n + 1) - --- F(n + 1) 319 638 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1103 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 j) F(n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 537 8 63 2 9 663 3 G(n) = --- F(n + 1) + -- F(n) F(n + 1) - --- F(n) F(n + 1) 638 44 638 2685 2 6 696343 6 5 1396053 4 3 - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) - ------- F(n) F(n + 1) 638 116 1276 27 4 531 3 1743055 3 4 1750427 3 8 - -- F(n) + --- F(n) + ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 29 58 638 638 32399 6 158127 5 6 537 5 3 - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + --- F(n) F(n + 1) 319 638 319 120074 5 2 43 7 8390473 4 7 + ------ F(n) F(n + 1) - ---- F(n + 1) + ------- F(n) F(n + 1) 319 1276 1276 2685 4 4 23 3 45 1137 4 + ---- F(n) F(n + 1) + --- F(n + 1) + -- - ---- F(n + 1) 638 116 58 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1104 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 25435725 3 8 48948409 3 4 G(n) = -------- F(n) F(n + 1) + -------- F(n) F(n + 1) 551 6061 1403333 2 67354975 10 - ------- F(n) F(n + 1) - -------- F(n) F(n + 1) 12122 1102 7518567 2 2118 615 2 + ------- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n + 1) 12122 6061 12122 4590317 2 5 13409200 11 74310 3 1578 2 - ------- F(n) F(n + 1) + -------- F(n + 1) + ----- F(n) - ---- F(n) 638 551 6061 6061 265180519 7 1355568 3 83732871 6 - --------- F(n + 1) - ------- F(n + 1) + -------- F(n) F(n + 1) 12122 551 12122 7330925 2 9 + ------- F(n) F(n + 1) 1102 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1105 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 j) F(n + 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 8 3 G(n) = -1/3080 F(n) (1407681 F(n) F(n + 1) - 4766905 F(n) F(n + 1) 4 7 5 6 6 3 - 1829806 F(n) F(n + 1) + 4026330 F(n) F(n + 1) - 3750 F(n) F(n + 1) 7 9 11 + 3245 F(n + 1) + 750 F(n + 1) - 2065 F(n + 1) 5 2 4 9 2 + 688512 F(n) F(n + 1) - 600 F(n) F(n + 1) + 732138 F(n) F(n + 1) 10 2 2 7 - 312705 F(n) F(n + 1) - 66935 F(n) F(n + 1) - 4500 F(n) F(n + 1) 2 5 2 3 5 + 65670 F(n) F(n + 1) + 750 F(n) F(n + 1) - 1350 F(n + 1) 4 5 11 9 5 + 8100 F(n) F(n + 1) + 55140 F(n) - 300 F(n) + 600 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1106 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 j) F(n + 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2637 7 12431 8 15027 7 28577 4 G(n) = ---- F(n) + ----- F(n + 1) + ----- F(n + 1) - ----- F(n + 1) 6061 96976 12122 48488 8487 3 5925 11 19077 4 3 39803 - ----- F(n + 1) - ----- F(n + 1) + ----- F(n) F(n + 1) + ----- 12122 12122 12122 96976 34001875 4 8 2733 6 9407801 3 + -------- F(n) F(n + 1) + ----- F(n) F(n + 1) + ------- F(n) F(n + 1) 1102 12122 48488 16377 3 4 603178245 3 5 + ----- F(n) F(n + 1) + --------- F(n) F(n + 1) 6061 48488 107693605 6 2 29625 6 5 - --------- F(n) F(n + 1) - ----- F(n) F(n + 1) 96976 12122 14454750 6 6 31632490 4 4 - -------- F(n) F(n + 1) - -------- F(n) F(n + 1) 551 6061 7275 5 2 57371353 5 3 35550 5 6 - ---- F(n) F(n + 1) + -------- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 24244 6061 18625 5 7 29625 3 8 6959625 3 9 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ------- F(n) F(n + 1) 29 6061 551 1827563 4 - ------- F(n) 96976 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1107 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 j) F(n + 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 223071 187757849 4 4875 3 6219 2 G(n) = - ------ - --------- F(n + 1) + ----- F(n) - ----- F(n) F(n + 1) 12122 12122 12122 12122 720791 2 2 17329455 2 6 - ------ F(n) F(n + 1) - -------- F(n) F(n + 1) 638 551 34001875 2 10 1402739 3 + -------- F(n) F(n + 1) + ------- F(n) F(n + 1) 1102 6061 951229 3 8937805 7 + ------ F(n) F(n + 1) + ------- F(n) F(n + 1) 209 551 137685750 11 25246695 3 5 - --------- F(n) F(n + 1) + -------- F(n) F(n + 1) 551 551 101816625 3 9 4374 2 690 3 + --------- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n + 1) 551 6061 6061 109130125 12 92040825 8 + --------- F(n + 1) - -------- F(n + 1) 1102 1102 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1108 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 7635225 11 205747 2 5 3921 3 93 G(n) = ------- F(n + 1) - ------ F(n) F(n + 1) + ---- F(n) - --- 319 29 319 319 1560853 3 306 3 81 4 - ------- F(n + 1) + --- F(n) F(n + 1) + --- F(n + 1) 638 319 638 395293 2 73933 2 19179400 10 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) - -------- F(n) F(n + 1) 638 638 319 195484 6 27 2 2 14474300 3 8 + ------ F(n) F(n + 1) - -- F(n) F(n + 1) + -------- F(n) F(n + 1) 29 29 319 2555452 3 4 2098075 2 9 144 3 + ------- F(n) F(n + 1) + ------- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 319 6854746 7 - ------- F(n + 1) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1109 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(4 j) F(2 n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 1143 5 31055125 12 26148520 8 5865 G(n) = ---- F(n + 1) + -------- F(n + 1) - -------- F(n + 1) - ---- 638 319 319 319 123 624 9711125 2 10 + --- F(n) - --- F(n + 1) + ------- F(n) F(n + 1) 319 319 319 898430 2 6 747 2 3 14472950 3 5 - ------ F(n) F(n + 1) - --- F(n) F(n + 1) + -------- F(n) F(n + 1) 29 319 319 3429 3 2 13453 3 2637 4 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 58 638 1443358 3 57919375 3 9 + ------- F(n) F(n + 1) + -------- F(n) F(n + 1) 319 319 718523 2 2 78373250 11 - ------ F(n) F(n + 1) - -------- F(n) F(n + 1) 638 319 172230 7 9801375 4 + ------ F(n) F(n + 1) - ------- F(n + 1) 11 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1110 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 7 10 6 695 3 G(n) = -9/44 F(n) + 3/44 F(n) + -- F(n) F(n + 1) + --- F(n) F(n + 1) 11 22 1028 4 4 4127 3 5 25 3 4 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - -- F(n) F(n + 1) 11 110 44 747 3 720 2 6 359 7 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 110 11 11 15 4 3 5 2 588 8 971 4 + -- F(n) F(n + 1) + 5/44 F(n) F(n + 1) - --- F(n) + --- F(n) 22 55 110 2 5 7 8 - 9/11 F(n) F(n + 1) - 5/22 F(n + 1) + 5/22 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1111 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 8 2 3 G(n) = 1/12 F(n) F(n + 1) - 14605/4 F(n) F(n + 1) - 1975/6 F(n) F(n + 1) 3 6 9 + 17665/6 F(n) F(n + 1) + 3 F(n) - 131/4 F(n + 1) + 8855/6 F(n + 1) 2 3 2 2695 2 7 - 1/12 F(n) F(n + 1) + 177 F(n) F(n + 1) + ---- F(n) F(n + 1) 12 4 17317 5 + 1895/3 F(n) F(n + 1) - ----- F(n + 1) 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1112 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 8735 7 2 10 7 20 4 4 G(n) = ---- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 88 11 11 906 6 3 24 6 2 19 2 2 - --- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 11 11 22 89 8 7 3 - -- F(n) F(n + 1) + 2/11 F(n) F(n + 1) - 4/11 F(n) F(n + 1) 88 997 4 5 4 1175 4 5 - --- F(n) F(n + 1) - 25 F(n) F(n + 1) + ---- F(n) F(n + 1) 44 44 5 4 71 4 9 + 35/8 F(n) + 7/22 F(n) - 5/22 + -- F(n) F(n + 1) + 5/22 F(n + 1) 44 129 - --- F(n) 88 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1113 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 445 3 G(n) = -1/12 F(n) F(n + 1) + 25/6 F(n) - --- F(n + 1) - 1/6 F(n) 12 2 9 3 2 + 1/12 F(n) F(n + 1) + 8855/6 F(n + 1) + 187 F(n) F(n + 1) 2695 2 7 4 8 + ---- F(n) F(n + 1) + 621 F(n) F(n + 1) - 14605/4 F(n) F(n + 1) 12 2 3 3 6 5 - 659/2 F(n) F(n + 1) + 17665/6 F(n) F(n + 1) - 5755/4 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1114 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n + j) F(n + 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1409761 5 427089 2266195 2 8 G(n) = ------- F(n) F(n + 1) + ------ F(n) F(n + 1) + ------- F(n) F(n + 1) 638 638 319 918385 9 491569 4 2 119990 3 7 - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 319 319 319 779655 3 3 34 6 85 5 2 + ------ F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 638 319 319 388527 5 1848270 4 6 28 3 - ------ F(n) F(n + 1) - ------- F(n) F(n + 1) - --- F(n) 638 319 319 107 2 85 3 4 1886 2 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) 319 319 319 17 6 20 10 20 3 + --- F(n) F(n + 1) - --- F(n + 1) + --- F(n + 1) 319 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1115 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n + j) F(n + 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 45 3 2 5 3 7 G(n) = --- F(n) F(n + 1) - 5/22 - 7/11 F(n) F(n + 1) + 7/22 F(n) F(n + 1) 22 63 4 4 4 5 375 4 - -- F(n) F(n + 1) + 3/2 F(n) F(n + 1) - --- F(n) F(n + 1) 22 22 18 2 2 15 8 13 3 - -- F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 11 11 11 40 2 7 43 9 8 30 8 + -- F(n) F(n + 1) + -- F(n) + 7/22 F(n) - -- F(n) F(n + 1) 11 11 11 49 7 1755 7 2 6 2 + -- F(n) F(n + 1) + ---- F(n) F(n + 1) + 7/2 F(n) F(n + 1) 22 22 1445 6 3 9 - ---- F(n) F(n + 1) + 5/22 F(n + 1) 22 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1116 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n + j) F(n + 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 3 3 7 G(n) = -1/3 F(n) F(n + 1) + 3961/3 F(n) F(n + 1) + 36325/3 F(n) F(n + 1) 2 4 5 9 - 6555/4 F(n) F(n + 1) + 14111/6 F(n) F(n + 1) - 46700/3 F(n) F(n + 1) 2 2 2 8 + 1/12 F(n) F(n + 1) + 5475/4 F(n) F(n + 1) + 187/3 F(n) F(n + 1) 3 4039 2 73 2 4 + 1/6 F(n) F(n + 1) + 1/12 - ---- F(n + 1) - -- F(n) - 1/12 F(n + 1) 12 12 6 74875 10 - 5903 F(n + 1) + ----- F(n + 1) 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1117 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n + j) F(n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 205 2 7 21749 2 4 158 2 3 G(n) = --- F(n) F(n + 1) - ----- F(n) F(n + 1) - --- F(n) F(n + 1) 462 4510 77 181 8 92589 10 10 - --- F(n) F(n + 1) - ----- F(n) + 5/22 F(n + 1) 462 4510 207 3 6 538004 7 3 25 33277 2 - --- F(n) F(n + 1) + ------ F(n) F(n + 1) - --- F(n) + ----- F(n) 308 2255 924 2255 9 185 9 613 3 2 57883 2 8 - 5/22 F(n + 1) - --- F(n) + --- F(n) F(n + 1) - ----- F(n) F(n + 1) 924 462 902 7811 53 4 614411 8 2 + ---- F(n) F(n + 1) + -- F(n) F(n + 1) - ------ F(n) F(n + 1) 2255 44 4510 116813 9 85 8 83027 5 - ------ F(n) F(n + 1) + --- F(n) F(n + 1) - ----- F(n) F(n + 1) 4510 231 2255 142847 9 + ------ F(n) F(n + 1) 4510 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1118 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n + j) F(n + 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 151121 2 7265 10 4527 2 3 G(n) = ------ F(n + 1) - ---- F(n) + ---- F(n) - 1/3 F(n) F(n + 1) + 1/12 156 78 52 107449 2 4 149725 2 8 46713 5 + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 156 156 26 3 93898 3 3 2 2 + 1/6 F(n) F(n + 1) - ----- F(n) F(n + 1) + 1/12 F(n) F(n + 1) 39 46750 9 4 46433 6 + ----- F(n) F(n + 1) - 1/12 F(n + 1) - ----- F(n + 1) 39 78 12112 58255 10 - ----- F(n) F(n + 1) - ----- F(n + 1) 39 156 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1119 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n + j) F(n + 3 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5551 3 796190 3 8 20 151 4 G(n) = ---- F(n) - ------ F(n) F(n + 1) + --- + --- F(n) 638 319 319 638 1512791 2 5 1502331 4 3 - ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 319 638 991583 3 4 64421 2 3 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) - 7/319 F(n) F(n + 1) 319 638 22 2 2 134790 10 9733320 2 9 - -- F(n) F(n + 1) - ------ F(n) F(n + 1) + ------- F(n) F(n + 1) 29 11 319 251 3 7448193 6 7635245 4 7 + --- F(n) F(n + 1) + ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 638 638 319 184785 2 20 11 + ------ F(n) F(n + 1) - --- F(n + 1) 319 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1120 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n + j) F(n + 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 6 4 2 4 G(n) = -1/12 F(n) (-F(n + 1) - 109 F(n) + F(n) - 35030 F(n) F(n + 1) 2 2 9 7 10 + F(n) F(n + 1) + 25 F(n) F(n + 1) + 4 F(n) F(n + 1) - 5 F(n + 1) 4 2 3 7 3 5 - 4603 F(n) F(n + 1) - 83600 F(n) F(n + 1) - 10 F(n) F(n + 1) 3 3 3 2 8 + 14102 F(n) F(n + 1) - 5 F(n) F(n + 1) + 34900 F(n) F(n + 1) 6 2 5 5 5 3 + 3 F(n) F(n + 1) + 77165 F(n) F(n + 1) + 8 F(n) F(n + 1) 5 4 6 + 1029 F(n) F(n + 1) - 3875 F(n) F(n + 1) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1121 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n + j) F(2 n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 22888 5 19 4 237125 3 7 G(n) = ----- F(n) F(n + 1) - -- F(n) F(n + 1) + ------ F(n) F(n + 1) 11 22 22 13685 3 3 23 3 2 33029 2 4 + ----- F(n) F(n + 1) + -- F(n) F(n + 1) - ----- F(n) F(n + 1) 11 11 22 2 3 306975 9 3537 2 - F(n) F(n + 1) - ------ F(n) F(n + 1) - ---- F(n + 1) 22 11 1335 28525 2 8 115771 6 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - ------ F(n + 1) 22 22 22 13 5 69 2 61425 10 + 3/11 F(n) - -- F(n + 1) + 4/11 F(n + 1) - -- F(n) + ----- F(n + 1) 22 11 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1122 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 588925 3 8 G(n) = -1/6 F(n) F(n + 1) + 600 F(n) F(n + 1) + ------ F(n) F(n + 1) 12 29543 3 7 3 11 - ----- F(n + 1) - 70108/3 F(n + 1) + 35/4 F(n) + 103325/4 F(n + 1) 12 98719 3 4 3 2 5 + ----- F(n) F(n + 1) - 7/12 F(n) F(n + 1) - 1/12 F(n + 1) + 1/4 F(n) 12 10 1241 2 + 1/12 F(n + 1) - 389075/6 F(n) F(n + 1) - ---- F(n) F(n + 1) 12 2 3 91913 6 83575 2 9 + 3/4 F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 12 12 91049 2 5 - ----- F(n) F(n + 1) 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1123 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n + j) F(2 n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 14638 2 101 3 25 6 4 G(n) = ------ F(n) F(n + 1) + --- F(n) - -- F(n) F(n + 1) 11 11 22 63975 6 5 52903 4 3 67 4 2 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - -- F(n) F(n + 1) 11 22 44 58875 3 8 25 3 7 66727 5 2 - ----- F(n) F(n + 1) + -- F(n) F(n + 1) + ----- F(n) F(n + 1) 22 11 22 32 5 70775 4 7 6 + -- F(n) F(n + 1) + ----- F(n) F(n + 1) + 1229 F(n) F(n + 1) 11 11 2080 5 6 30 5 5 6 + ---- F(n) F(n + 1) - -- F(n) F(n + 1) - 3/2 F(n) F(n + 1) 11 11 15 5 14671 2 5 2 4 + -- F(n) F(n + 1) + ----- F(n) F(n + 1) - 7/44 F(n) F(n + 1) 11 11 39 2 31 6 10 11 - -- F(n) + -- F(n) - 5/22 F(n + 1) + 5/22 F(n + 1) 44 44 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1124 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 3 G(n) = ) F(j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 357 2 1139 2 8 857297 6 4 G(n) = ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 638 319 638 15 6 60943 5 5 75 5 2 - --- F(n) F(n + 1) + ----- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 638 15933 5 416786 4 6 127929 4 2 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 319 319 638 181078 3 7 75 3 4 32937 3 3 - ------ F(n) F(n + 1) + --- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 638 58 185 3 15 6 205 3 5309 6 + --- F(n + 1) - --- F(n) F(n + 1) + --- F(n) - ---- F(n) 638 638 638 638 89 185 10 + -- F(n) F(n + 1) - --- F(n + 1) 58 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1125 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 2 3 G(n) = -1/12 F(n) (-4 F(n + 1) - 36200 F(n + 1) - 7798 F(n) F(n + 1) 8 2 9 - 91300 F(n) F(n + 1) + 4 F(n) F(n + 1) + 37050 F(n + 1) 2 4 3 2 + 6 F(n) F(n + 1) + 15398 F(n) F(n + 1) + 4267 F(n) F(n + 1) 2 7 3 6 3 + 4450 F(n) F(n + 1) + 74875 F(n) F(n + 1) - 3 F(n) - 844 F(n + 1) + 99 F(n)) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1126 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 15270635 4 7 1558533 4 3 G(n) = --------- F(n) F(n + 1) - ------- F(n) F(n + 1) 638 638 796335 3 8 94 3 19466785 2 9 - ------ F(n) F(n + 1) - --- F(n) F(n + 1) + -------- F(n) F(n + 1) 319 319 638 269941 2 5 222 2 2 37039 2 - ------ F(n) F(n + 1) + --- F(n) F(n + 1) - ----- F(n) F(n + 1) 58 319 319 134785 10 337472 6 63 3 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - --- F(n) F(n + 1) 11 29 319 394249 2 994889 3 4 185 11 + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - --- F(n + 1) 638 319 638 7871 3 200 4 185 + ---- F(n) - --- F(n) + --- 638 319 638 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1127 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 149 11 4 3 4 G(n) = --- F(n) + 22/5 F(n) F(n + 1) - 7/12 F(n) F(n + 1) 12 3 8 3 2 9151 5 2 + 17/6 F(n) F(n + 1) + 7/12 F(n) F(n + 1) + ---- F(n) F(n + 1) 60 551 4 7 21 2 841 10 - --- F(n) F(n + 1) - -- F(n) F(n + 1) - --- F(n) F(n + 1) 15 20 12 1091 9 2 8 3 7 4 + ---- F(n) F(n + 1) - 1255/2 F(n) F(n + 1) + 2109/4 F(n) F(n + 1) 30 7 5 3 + 3/20 F(n + 1) - F(n) + 7/12 F(n) - 3/20 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1128 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 14582963 8 4 6338 4 7 G(n) = --------- F(n) F(n + 1) + ---- F(n) F(n + 1) 12122 6061 13397237 9 3 7157 4 8 1381 2 9 + -------- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 209 6061 10080 3 9 1633 3 8 5837 9 2 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 12122 12122 6903 8 3 5003193 7 5 2732 7 4 + ---- F(n) F(n + 1) - ------- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 6061 6061 948494 7 2842283 6 6 12175 6 5 + ------ F(n) F(n + 1) + ------- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 12122 6061 8552359 6 2 2733 6 1110568 5 7 - ------- F(n) F(n + 1) + ----- F(n) F(n + 1) + ------- F(n) F(n + 1) 12122 12122 6061 30 5 6 2040 222591 8 315 7 - ---- F(n) F(n + 1) + ---- - ------ F(n) + ----- F(n) 6061 6061 12122 12122 283 3 1988 4 973 7 1018 8 - ---- F(n + 1) - ---- F(n + 1) + ----- F(n + 1) + ---- F(n + 1) 1102 6061 12122 6061 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1129 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n + 2 j) F(n + 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2098075 2 9 13710797 7 81 4 G(n) = ------- F(n) F(n + 1) - -------- F(n + 1) + --- F(n + 1) 319 638 638 1559693 3 197226 2 31 3 - ------- F(n + 1) + ------ F(n) F(n + 1) + --- F(n) F(n + 1) 638 319 638 4303983 6 7697 3 14474300 3 8 + ------- F(n) F(n + 1) + ---- F(n) + -------- F(n) F(n + 1) 638 638 319 25 2 2 4527159 2 5 19179400 10 - -- F(n) F(n + 1) - ------- F(n) F(n + 1) - -------- F(n) F(n + 1) 58 638 319 73527 2 2554437 3 4 13 3 - ----- F(n) F(n + 1) + ------- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 319 41 7635225 11 - --- + ------- F(n + 1) 638 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1130 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n + 2 j) F(n + 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 537 2 3 29903180 2 6 G(n) = ---- F(n) F(n + 1) + -------- F(n) F(n + 1) 638 6699 24693125 2 10 5278666 3 239 3 2 - -------- F(n) F(n + 1) - ------- F(n) F(n + 1) + --- F(n) F(n + 1) 2233 2233 638 15 73 345383405 8 79547945 3 5 - --- F(n) - --- F(n + 1) - --------- F(n + 1) - -------- F(n) F(n + 1) 638 319 13398 1218 79761568 3 85 4 301687125 7 - -------- F(n) F(n + 1) - --- F(n) F(n + 1) + --------- F(n) F(n + 1) 6699 638 4466 93 5 396621539 4 30581000 11 + --- F(n + 1) + --------- F(n + 1) + -------- F(n) F(n + 1) 319 13398 2233 11102177 47885225 2 2 113580125 12 + -------- + -------- F(n) F(n + 1) - --------- F(n + 1) 26796 8932 26796 11583875 12 - -------- F(n) 26796 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1131 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + 2 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5659 10 247 9 1111 9 67 5 G(n) = ---- F(n + 1) + --- F(n + 1) - ---- F(n) F(n + 1) - --- F(n + 1) 682 495 62 110 179 58 30257 5 4 - --- F(n) - --- F(n + 1) + ----- F(n) F(n + 1) + 4/3 F(n) F(n + 1) 330 495 341 87 85621 9 2296 8 2 - --- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 341 682 11 637 8 92748 7 3 1313 7 2 + --- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 495 341 990 1627 2 3 923 8 30137 2 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) + ----- F(n + 1) 495 990 682 33159 10 52 9 13929 2 35641 6 - ----- F(n) + --- F(n) + ----- F(n) - ----- F(n + 1) 682 165 341 682 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1132 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n + 2 j) F(2 n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 11 7 3 G(n) = -1/4 F(n) + -- F(n + 1) - 69854/3 F(n + 1) + 49/4 F(n) 12 11 30559 3 6 + 103325/4 F(n + 1) - ----- F(n + 1) + 29803/4 F(n) F(n + 1) 12 10 1421 2 2 3 - 389075/6 F(n) F(n + 1) - ---- F(n) F(n + 1) - 1/12 F(n) F(n + 1) 12 588925 3 8 90937 2 5 100787 3 4 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 12 12 12 2 4 11 5 + 639 F(n) F(n + 1) + 7/3 F(n) F(n + 1) - -- F(n + 1) 12 83575 2 9 3 2 + ----- F(n) F(n + 1) - 9/4 F(n) F(n + 1) 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1133 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n + 2 j) F(2 n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 257 11 12 6 2 551 10 G(n) = --- F(n) - -- F(n) F(n + 1) - 5/22 F(n + 1) - --- F(n) F(n + 1) 22 11 22 32 2 4 21 5 1977 9 2 - -- F(n) F(n + 1) + -- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 22 11 10935 8 3 6487 7 4 967 6 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 22 22 22 959 5 2 5 86 4 7 + --- F(n) F(n + 1) + 3/22 F(n) F(n + 1) - -- F(n) F(n + 1) 11 11 53 4 2 24 3 8 71 3 3 - -- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 22 48 2 5 6 3 + -- F(n) F(n + 1) + 7/22 F(n) + 5/22 F(n + 1) 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1134 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 101 9 4135 2187 8 5157285 5 7 G(n) = --- F(n) - ---- - ---- F(n + 1) - ------- F(n) F(n + 1) 638 5104 5104 2552 61405 6 2 425 6 3 23816365 6 6 - ----- F(n) F(n + 1) + --- F(n) F(n + 1) + -------- F(n) F(n + 1) 88 638 5104 206725 7 189 7 2 3417285 7 5 + ------ F(n) F(n + 1) - --- F(n) F(n + 1) + ------- F(n) F(n + 1) 1276 58 2552 545 8 28398245 8 4 424 2 7 - --- F(n) F(n + 1) - -------- F(n) F(n + 1) + --- F(n) F(n + 1) 638 5104 319 1090 3 6 14265 3 9 57 4 5 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - -- F(n) F(n + 1) 319 1276 58 282365 4 8 694536 5 3 3775 5 4 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 5104 319 638 155 2421 4 125 5 88545 8 + --- F(n + 1) + ---- F(n + 1) - --- F(n + 1) - ----- F(n) 319 2552 638 5104 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1135 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n + 2 j) F(2 n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 8 2 9 G(n) = -1/12 F(n) (1262875 F(n) F(n + 1) + 166250 F(n) F(n + 1) 3 2 3 4 2 - 53 F(n) F(n + 1) + 193375 F(n) F(n + 1) - 2390 F(n) F(n + 1) 2 3 2 5 10 + 8 F(n) F(n + 1) - 186910 F(n) F(n + 1) - 1652625 F(n) F(n + 1) 4 6 11 + 59 F(n) F(n + 1) + 205465 F(n) F(n + 1) + 659750 F(n + 1) 2 7 3 + 13728 F(n) F(n + 1) + 20 F(n + 1) - 602790 F(n + 1) - 56952 F(n + 1) 5 3 - 26 F(n + 1) - 4 F(n) + 220 F(n) ) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1136 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 730 10 802331 11 2 2 G(n) = ---- F(n) F(n + 1) + ------ F(n) F(n + 1) - 18659/6 F(n) F(n + 1) 2607 132 12 11 7301 3 4 + 5/22 F(n + 1) - 5/22 F(n + 1) - ----- F(n) - 3123/2 F(n) 10428 1613 11 33969 12 545 3 8 + ----- F(n) + ----- F(n) + ----- F(n) F(n + 1) 10428 22 10428 65245 3 9 103735 3 4 27811 2 5 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 12 10428 5214 123661 3 5 2 6 3491 3 - ------ F(n) F(n + 1) + 16457/3 F(n) F(n + 1) + ---- F(n) F(n + 1) 12 12 1275 2 9 2 10 37147 7 - ---- F(n) F(n + 1) - 8558 F(n) F(n + 1) - ----- F(n) F(n + 1) 1738 12 8303 10 408311 11 14449 6 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 10428 132 10428 8639 2 10 2 14625 4 3 + ---- F(n) F(n + 1) + 14105/3 F(n) F(n + 1) - ----- F(n) F(n + 1) 5214 1738 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1137 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(n + 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 4 4 G(n) = -1/12 F(n) (-149 F(n) - 39689 F(n) F(n + 1) - 6 F(n + 1) 6 2 10 - 150567 F(n + 1) - 7638 F(n + 1) + 158200 F(n + 1) 3 2 2 + 1411 F(n) F(n + 1) + 12 F(n) F(n + 1) + 5 F(n) F(n + 1) 3 3 9 2 8 + 30499 F(n) F(n + 1) - 393550 F(n) F(n + 1) + 31025 F(n) F(n + 1) 3 5 3 7 - 11 F(n) F(n + 1) + 60478 F(n) F(n + 1) + 309975 F(n) F(n + 1) + 5) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1138 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n + 3 j) F(2 n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5584 8 20 4190875 5 7 30 8 G(n) = ----- F(n) - --- - ------- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 319 319 3271816 3 25 4 23725503 7 - ------- F(n) F(n + 1) - -- F(n) F(n + 1) + -------- F(n) F(n + 1) 319 29 319 360 8 20453875 11 325279 2 2 + --- F(n) F(n + 1) - -------- F(n) F(n + 1) - ------ F(n) F(n + 1) 319 319 319 248 2 3 14457541 2 6 970 2 7 - --- F(n) F(n + 1) - -------- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 319 820 4 5 39436875 4 8 100 5 4 + --- F(n) F(n + 1) - -------- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 4468000 2 10 50713 3 211 3 2 + ------- F(n) F(n + 1) + ----- F(n) F(n + 1) + --- F(n) F(n + 1) 29 319 319 4546891 4 4 20 9 424628 5 3 + ------- F(n) F(n + 1) + --- F(n + 1) + ------ F(n) F(n + 1) 319 319 29 138 5 - --- F(n) 319 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1139 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(n + 3 j) F(2 n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 8 2 3 7 G(n) = -1/2 F(n) F(n + 1) - 1/12 F(n + 1) + 3/2 F(n) F(n + 1) 4 2 4 17233 8 4 - 1/2 F(n) F(n + 1) + 1/2 F(n + 1) - ----- F(n) F(n + 1) 12 7 5 5 3 7 3 + 2923/3 F(n) F(n + 1) + 706/3 F(n) F(n + 1) - 11 F(n) F(n + 1) 8 2 895 4 8 5 + 1/2 F(n) F(n + 1) - 1/3 - --- F(n) F(n + 1) - 7/2 F(n) F(n + 1) 12 11 10 2 9 3 + 488/3 F(n) F(n + 1) - 1054/3 F(n) F(n + 1) + 1627/3 F(n) F(n + 1) 9 3 9 6 2 + 25/2 F(n) F(n + 1) + 37/3 F(n) F(n + 1) - 139/3 F(n) F(n + 1) 8 10 6 6 8 - 109/6 F(n) + 5 F(n) - 9/2 F(n) + 1/12 F(n + 1) - 1/6 F(n + 1) Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1140 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 304 238546 2 5 5946 2 G(n) = --- F(n) F(n + 1) - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 165 775 775 25127 10 481 9 134 5 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 1550 165 55 115276 2 1027 2 8 3149 8 2 - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 775 165 330 70249 10 1576 9 235493 8 3 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 155 165 310 1816 3 3 2 4 10 102279 3 + ---- F(n) F(n + 1) + F(n) F(n + 1) - 5/22 F(n + 1) - ------ F(n) 165 775 96 2 267 10 11 2466363 11 + -- F(n) - --- F(n) + 5/22 F(n + 1) + ------- F(n) 55 110 17050 126677 6 2329 2 9 9 2 + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) + 558 F(n) F(n + 1) 775 62 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1141 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ 2 G(n) = ) F(j) F(2 n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 4 29 6 1262875 12 G(n) = F(n) F(n + 1) - 64313/4 F(n + 1) - -- F(n + 1) + ------- F(n + 1) 12 12 3 2 10 - 18 + 14096/3 F(n) F(n + 1) + 194875/6 F(n) F(n + 1) 3 3 3 3 5 + 233 F(n) F(n + 1) - 6 F(n) F(n + 1) + 47775 F(n) F(n + 1) 2 6 7 - 99745/3 F(n) F(n + 1) - 1/2 F(n) F(n + 1) + 54155/3 F(n) F(n + 1) 2 2 5 11 - 6935/6 F(n) F(n + 1) + 11/2 F(n) F(n + 1) - 796375/3 F(n) F(n + 1) 3 9 29 2 8 + 196625 F(n) F(n + 1) + -- F(n + 1) - 267430/3 F(n + 1) 12 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ Theorem Number, 1142 Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- \ n t ) F(n) t = ---------- / 2 ----- 1 - t - t n = 0 Also Let G(n) be the sequence defined by n - 1 ----- \ G(n) = ) F(j) F(2 n + j) F(2 n + 2 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 21 7 417 8 12 G(n) = -5/22 F(n + 1) + -- F(n) - --- F(n) + 5/22 F(n + 1) 22 22 12 10 463 7 189 6 + -- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 55 50 55 697 3 267 2 1179 2 6 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 50 110 25 120 8 3 923 2 5 87501 7 5 + --- F(n) F(n + 1) - --- F(n) F(n + 1) - ----- F(n) F(n + 1) 11 110 110 12 7 4 3511 7 104719 6 6 + -- F(n) F(n + 1) + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) 11 22 275 456 6 5 393281 6 2 211 6 - --- F(n) F(n + 1) - ------ F(n) F(n + 1) - --- F(n) F(n + 1) 55 550 110 871 2 2 2 787 11 + --- F(n) F(n + 1) + 8/55 F(n) F(n + 1) + --- F(n) F(n + 1) 50 275 8 4 117373 9 3 48 9 2 - 2227/2 F(n) F(n + 1) + ------ F(n) F(n + 1) + -- F(n) F(n + 1) 55 11 Proof: Both sides are C-finite, so it is enough to check the first few terms, and this is left to the dear reader. ------------------------------------------------ I hope that you enjoyed, dear reader, these, 1142, beautiful and deep theorems. 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