------------------------------ 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 ----- \ 2 G(n) = ) F(j) F(n - j) / ----- j = 0 then we have the following closed-form expression for G(n) G(n) = -1/2 (F(n) - 1 + 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, 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 ----- \ 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 2 G(n) = -1/6 (F(n) - 2 F(n + 1)) (F(n) - F(n) F(n + 1) - 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, 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 ----- \ 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 3 2 G(n) = -1/2 F(n + 1) + 1/2 F(n + 1) + 1/2 F(n) F(n + 1) - 1/2 F(n) 2 3 - 1/2 F(n) F(n + 1) + 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, 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 ----- \ 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) 17 3 G(n) = - 1/11 + 1/11 F(n + 1) - 5/22 F(n) + -- F(n) F(n + 1) 22 29 2 2 3 4 - -- F(n) F(n + 1) + 5/11 F(n) F(n + 1) + 7/22 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, 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 ----- \ 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 2 G(n) = 1/3 - 1/3 F(n + 1) + 1/3 F(n) F(n + 1) - 1/6 F(n) 2 2 3 4 + 11/6 F(n) F(n + 1) - 11/6 F(n) F(n + 1) - 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 ----- \ 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 2 3 G(n) = 1/2 - 1/2 F(n + 1) + 1/2 F(n) F(n + 1) + F(n) F(n + 1) 2 2 2 3 4 - 3/2 F(n) F(n + 1) + 1/2 F(n) F(n + 1) + F(n) - 3/2 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, 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 ----- \ 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) 2 G(n) = 1/2 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, 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 ----- \ 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 25 13 G(n) = -1/11 F(n + 1) + 1/11 F(n + 1) + -- F(n) - -- F(n) F(n + 1) 22 22 2 2 3 3 2 + 9/22 F(n) - 4/11 F(n) F(n + 1) + 7/2 F(n) F(n + 1) 28 4 17 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, 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 ----- \ 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) 4 3 2 G(n) = -1/6 (-2 F(n + 1) + F(n)) (F(n) - 9 F(n) F(n + 1) - 2 F(n) 2 2 2 + 9 F(n + 1) F(n) + F(n) F(n + 1) + 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, 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 ----- \ G(n) = ) F(j) F(n - j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 2 G(n) = - 1/2 + 1/2 F(n + 1) + F(n) - 1/2 F(n) F(n + 1) - 7/2 F(n) F(n + 1) 2 3 3 3 2 4 + 7/2 F(n) F(n + 1) + 9/2 F(n) F(n + 1) + F(n) F(n + 1) - F(n) 4 5 - 11/2 F(n) F(n + 1) + 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, 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 ----- \ G(n) = ) F(j) F(n - j) F(5 n - 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 75 2225 5 1625 2 4 G(n) = --- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 638 1850 3 3 590 2 1175 6 + ---- F(n) F(n + 1) + 5/638 F(n + 1) - --- F(n + 1) + ---- F(n + 1) 319 319 638 15 15 2 - --- 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, 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 ----- \ G(n) = ) F(j) F(n - j) F(5 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) G(n) = -1/2 F(n + 1) 5 4 4 5 (3 F(n + 1) - 3 F(n + 1) + 6 F(n + 1) F(n) - 8 F(n + 1) 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, 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 ----- \ G(n) = ) F(j) F(n - j) F(5 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 10 5 185 3 3 21 2 G(n) = -- F(n) F(n + 1) + --- F(n) F(n + 1) - -- F(n) F(n + 1) 11 22 22 14 2 10 6 27 + -- F(n) F(n + 1) + -- F(n + 1) - -- F(n) F(n + 1) 11 11 22 185 2 4 3 3 2 - --- F(n) F(n + 1) + 1/11 F(n + 1) - 7/11 F(n) - F(n + 1) 22 2 + 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, 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 ----- \ G(n) = ) F(j) F(n - j) F(5 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 2 G(n) = -9/2 F(n) F(n + 1) - 1/3 F(n) F(n + 1) - 5/2 F(n) F(n + 1) 2 4 3 4 + 25/2 F(n) F(n + 1) + 8/3 F(n) F(n + 1) + 1/6 F(n + 1) - 1/2 5 3 3 2 2 + 75/2 F(n) F(n + 1) - 275/6 F(n) F(n + 1) + 17 F(n + 1) + 1/2 F(n) 6 - 50/3 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, 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(5 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 5 G(n) = -5/2 F(n) - 12 F(n + 1) - 59/2 F(n + 1) + 23/2 F(n + 1) 6 4 + 30 F(n + 1) + 5/2 F(n) + 11 F(n) F(n + 1) - 26 F(n) F(n + 1) 5 2 3 2 4 - 70 F(n) F(n + 1) - 12 F(n) F(n + 1) - 5/2 F(n) F(n + 1) 3 2 3 3 + 69/2 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, 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(6 n - 6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 241 2 3 5 G(n) = ---- F(n) F(n + 1) - 1/21 F(n + 1) + 1/21 F(n + 1) + 1/42 F(n) 84 235 6 110 2 5 365 4 3 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 84 21 42 3 4 23 2 - 1/42 F(n) + 1/42 F(n) F(n + 1) + -- F(n) F(n + 1) 42 2 3 3 2 55 3 4 - 1/21 F(n) F(n + 1) - 1/42 F(n) F(n + 1) - -- F(n) F(n + 1) 14 4 + 1/21 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, 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(6 n - 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 53 2 2 7185 3 53 3 G(n) = --- F(n) + 5/638 F(n + 1) - ---- F(n + 1) - --- F(n) 638 638 638 1011 2 3590 7 16045 6 + ---- F(n) F(n + 1) + ---- F(n + 1) - ----- F(n) F(n + 1) 319 319 638 1665 3 4 46 235 2 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 58 319 638 3965 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, 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(6 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 3 3 2 G(n) = -1/2 F(n + 1) (-119 F(n) F(n + 1) + 123 F(n) F(n + 1) - 51 F(n + 1) 2 2 4 6 + 17 F(n) F(n + 1) - 3 F(n) - 18 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, 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(6 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 345 3 4 16 4 27 3 32 7 G(n) = --- F(n) F(n + 1) + -- F(n + 1) + -- F(n + 1) - -- F(n + 1) 22 11 55 55 23 7 186 5 2 333 2 15 53 4 - -- F(n) + --- F(n) F(n + 1) + --- F(n) F(n + 1) - -- + -- F(n) 22 55 110 11 22 89 3 27 2 2 232 4 3 - -- F(n) F(n + 1) + -- F(n) F(n + 1) - --- F(n) F(n + 1) 22 22 11 6 + 5/11 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, 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(6 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 2 3 G(n) = -29/3 F(n) F(n + 1) + 23/3 F(n) F(n + 1) - 41/6 F(n) F(n + 1) 2 5 3 2 5 + 80/3 F(n) F(n + 1) + 91/6 F(n) F(n + 1) + 9/2 F(n + 1) 2 6 3 4 - 193/6 F(n) F(n + 1) + 640/3 F(n) F(n + 1) - 1285/6 F(n) F(n + 1) 7 3 3 + 5/6 F(n) - 29/6 F(n + 1) - 545/6 F(n + 1) + 547/6 F(n + 1) - 5/6 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, 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 ----- \ G(n) = ) F(j) F(n - j) F(6 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 2 4 G(n) = 22 F(n) F(n + 1) + 1/2 F(n + 1) + 4 F(n) + 40 F(n) F(n + 1) 4 2 2 6 - 135/2 F(n) F(n + 1) + 117/2 F(n) F(n + 1) - 115/2 F(n) F(n + 1) 2 2 5 3 3 - 41/2 F(n) F(n + 1) + 285/2 F(n) F(n + 1) + 30 F(n) F(n + 1) 3 4 4 3 2 2 + 10 F(n) F(n + 1) - 135 F(n) F(n + 1) - 1/2 F(n + 1) - 4 F(n) 5 - 45/2 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, 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 ----- \ 2 G(n) = ) F(j) F(n - j) / ----- j = 0 then we have the following closed-form expression for G(n) G(n) = -1/2 (F(n) - 1 + 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, 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 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = -1/6 (F(n) - 2 F(n + 1)) (F(n) - F(n) F(n + 1) - 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, 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(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 17 3 G(n) = - 1/11 + 1/11 F(n + 1) - 5/22 F(n) + -- F(n) F(n + 1) 22 29 2 2 3 4 - -- F(n) F(n + 1) + 5/11 F(n) F(n + 1) + 7/22 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(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 G(n) = 1/2 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, 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(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 285 2 25 6 15 5 15 6 G(n) = 5/638 F(n + 1) - ---- F(n + 1) + --- F(n + 1) - --- F(n) + --- F(n) 1276 116 319 319 15 4 30 2 3 1715 2 4 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 1276 15 3 2 1415 3 3 30 4 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 319 638 319 1535 4 2 105 5 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 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, 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(6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 241 2 565 6 895 3 4 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 84 28 42 365 7 23 2 + 1/42 F(n) - 1/21 F(n + 1) - --- F(n + 1) + -- F(n) F(n + 1) 42 42 145 2 5 367 3 3 + --- F(n) F(n + 1) + --- F(n + 1) - 1/42 F(n) 42 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, 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 ----- \ G(n) = ) F(j) F(n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 2 G(n) = -1/2 F(n + 1) + 1/2 F(n + 1) + 1/2 F(n) F(n + 1) - 1/2 F(n) 2 3 - 1/2 F(n) F(n + 1) + 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, 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(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = 1/3 - 1/3 F(n + 1) + 1/3 F(n) F(n + 1) - 1/6 F(n) 2 2 3 4 + 11/6 F(n) F(n + 1) - 11/6 F(n) F(n + 1) - 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, 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(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 25 13 G(n) = -1/11 F(n + 1) + 1/11 F(n + 1) + -- F(n) - -- F(n) F(n + 1) 22 22 2 2 3 3 2 + 9/22 F(n) - 4/11 F(n) F(n + 1) + 7/2 F(n) F(n + 1) 28 4 17 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, 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(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) G(n) = -1/2 2 2 2 F(n + 1) F(n) (-F(n + 1) + F(n)) (3 F(n + 1) - 3 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, 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(n + 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 16045 6 53 2 53 3 46 G(n) = ------ F(n) F(n + 1) + --- F(n) - --- F(n) - --- F(n) F(n + 1) 638 638 638 319 1665 3 4 235 2 3965 2 5 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 58 638 638 2 7185 3 3590 7 1011 2 + 5/638 F(n + 1) - ---- F(n + 1) + ---- 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, 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) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 3 G(n) = 1/2 - 1/2 F(n + 1) + 1/2 F(n) F(n + 1) + F(n) F(n + 1) 2 2 2 3 4 - 3/2 F(n) F(n + 1) + 1/2 F(n) F(n + 1) + F(n) - 3/2 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, 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 + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 2 G(n) = -1/6 (-2 F(n + 1) + F(n)) (F(n) - 9 F(n) F(n + 1) - 2 F(n) 2 2 2 + 9 F(n) F(n + 1) + F(n) F(n + 1) + 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, 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 + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 27 185 2 4 21 2 G(n) = --- F(n) F(n + 1) - --- F(n) F(n + 1) - -- F(n) F(n + 1) 22 22 22 14 2 10 5 185 3 3 + -- F(n) F(n + 1) + -- F(n) F(n + 1) + --- F(n) F(n + 1) 11 11 22 10 6 2 2 3 3 + -- F(n + 1) - F(n + 1) + 7/11 F(n) + 1/11 F(n + 1) - 7/11 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, 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 + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 2 4 G(n) = 1/10 F(n + 1) (2 F(n + 1) F(n) - 50 F(n + 1) F(n) + 90 F(n + 1) F(n) 6 2 5 6 + 15 F(n) - 24 F(n + 1) - 57 F(n + 1) F(n) + 24 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, 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(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 2 G(n) = - 1/2 + 1/2 F(n + 1) + F(n) - 1/2 F(n) F(n + 1) - 7/2 F(n) F(n + 1) 2 3 3 3 2 4 + 7/2 F(n) F(n + 1) + 9/2 F(n) F(n + 1) + F(n) F(n + 1) - F(n) 4 5 - 11/2 F(n) F(n + 1) + 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, 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(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = - 1/2 + 17 F(n + 1) + 1/2 F(n) - 9/2 F(n) F(n + 1) 3 2 2 2 4 - 1/3 F(n) F(n + 1) - 5/2 F(n) F(n + 1) + 25/2 F(n) F(n + 1) 3 6 5 + 8/3 F(n) F(n + 1) - 50/3 F(n + 1) + 75/2 F(n) F(n + 1) 3 3 4 - 275/6 F(n) 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, 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(3 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 21 4 3 23 3 2 G(n) = -- F(n) + 1/11 - 1/11 F(n + 1) - -- F(n) + 3/11 F(n) F(n + 1) 22 22 35 6 95 2 5 160 4 3 + -- F(n) F(n + 1) - -- F(n) F(n + 1) - --- F(n) F(n + 1) 22 11 11 25 3 28 2 59 2 2 - -- F(n) F(n + 1) + -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 22 32 3 445 3 4 - -- 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, 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 ----- \ G(n) = ) F(j) F(n - j) F(4 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 5 2 3 G(n) = -1/2 F(n) F(n + 1) + F(n) F(n + 1) - 11/2 F(n) F(n + 1) 2 4 3 2 3 3 + 11/2 F(n) F(n + 1) + 9 F(n) F(n + 1) - 6 F(n) F(n + 1) 4 4 2 5 2 - 13/2 F(n) F(n + 1) - 8 F(n) F(n + 1) + 11 F(n) F(n + 1) - 5/2 F(n) 5 2 + 5/2 F(n) - 1/2 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, 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(4 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 3 4 G(n) = 9/2 F(n + 1) + 547/6 F(n + 1) - 29/3 F(n) F(n + 1) 2 2 3 2 5 + 23/3 F(n) F(n + 1) - 41/6 F(n) F(n + 1) + 80/3 F(n) F(n + 1) 3 2 7 2 + 91/6 F(n) F(n + 1) - 545/6 F(n + 1) - 193/6 F(n) F(n + 1) 3 4 6 - 1285/6 F(n) F(n + 1) + 640/3 F(n) F(n + 1) + 5/6 F(n) - 29/6 F(n + 1) 3 - 5/6 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, 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(5 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 5 3 3 G(n) = 135/2 F(n + 1) - 315/2 F(n) F(n + 1) + 165 F(n) F(n + 1) 2 7 2 2 + 117/2 F(n) F(n + 1) + 135 F(n + 1) - 68 F(n + 1) - 4 F(n) 6 2 2 4 - 655/2 F(n) F(n + 1) - 41/2 F(n) F(n + 1) - 55/2 F(n) F(n + 1) 2 5 3 4 + 15/2 F(n) F(n + 1) + 280 F(n) F(n + 1) + 22 F(n) F(n + 1) 3 3 - 269/2 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, 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 ----- \ 2 G(n) = ) F(j) F(2 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 G(n) = 9/22 - 9/22 F(n + 1) - 5/22 F(n) + 3/11 F(n) F(n + 1) 2 2 3 4 - 7/22 F(n) F(n + 1) + 5/11 F(n) F(n + 1) - 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, 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(2 n - 2 j) F(2 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = 1/3 - 1/3 F(n + 1) + 1/3 F(n) F(n + 1) - 2/3 F(n) 2 2 3 4 + 4/3 F(n) F(n + 1) - 4/3 F(n) 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, 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 ----- \ 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 G(n) = 9/22 F(n + 1) - 9/22 F(n + 1) + 3/22 F(n) - 1/11 F(n) F(n + 1) 2 2 3 3 2 - 1/11 F(n) + 3/22 F(n) F(n + 1) + 3/2 F(n) F(n + 1) 17 4 5 - -- F(n) F(n + 1) - 1/22 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, 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(2 n - 2 j) F(3 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 2 G(n) = -1/6 (-2 F(n + 1) + F(n)) (4 F(n) - 6 F(n) F(n + 1) - 5 F(n) 2 2 2 + 6 F(n + 1) F(n) + F(n) F(n + 1) + 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, 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(2 n - 2 j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 57 4 80 5 57 2 3 G(n) = --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 319 205 2 4 57 3 2 310 3 3 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 638 319 57 4 655 4 2 20 5 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) F(n + 1) 319 319 29 63 63 2 57 2 57 5 + --- 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, 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(2 n - 2 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 210 5 230 3 3 2 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) + 1/22 F(n) F(n + 1) 11 11 2 3 185 6 2 - 8/11 F(n) F(n + 1) - 9/22 F(n + 1) + --- F(n + 1) - 8 F(n + 1) 22 2 25 75 2 4 3 + 3/22 F(n) + -- F(n) F(n + 1) - -- F(n) F(n + 1) - 3/22 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, 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(2 n - 2 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 4 5 G(n) = -7/3 F(n) F(n + 1) - 3/2 + 7/6 F(n + 1) + 50 F(n) F(n + 1) 3 3 6 - 160/3 F(n) F(n + 1) - 65/3 F(n + 1) - 7 F(n) F(n + 1) 2 2 2 4 3 - 5 F(n) F(n + 1) + 10 F(n) F(n + 1) + 37/6 F(n) F(n + 1) 2 2 + 22 F(n + 1) + 3/2 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, 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(2 n - 2 j) F(5 n - 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 325 7 965 3 775 6 G(n) = 5/84 F(n) - --- F(n + 1) + --- F(n + 1) + --- F(n) F(n + 1) 28 84 28 725 3 4 2 25 2 5 - --- F(n) F(n + 1) + 5/7 F(n) F(n + 1) + -- F(n) F(n + 1) 28 14 2 - 5/84 F(n) + 5/42 F(n + 1) - 30/7 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, 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(2 n - 2 j) F(5 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 169 1047 2 750 2 5 G(n) = --- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 638 638 319 13025 6 12275 3 4 63 2 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + --- F(n + 1) 319 319 638 2 1939 2 5475 7 11013 3 - 5/638 F(n) + ---- F(n) F(n + 1) + ---- F(n + 1) - ----- F(n + 1) 319 319 638 3 + 5/638 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, 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(2 n - 2 j) F(5 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 175 6 400 2 5 G(n) = 9/22 F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 11 11 925 4 3 347 2 3 4 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - 1/22 F(n) + 1/22 F(n) 22 22 3 65 2 21 2 2 + 2/11 F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 3 225 3 4 4 + 3/11 F(n) F(n + 1) + --- F(n) F(n + 1) - 9/22 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, 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(2 n - 2 j) F(5 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 2 G(n) = 1/3 F(n + 1) + 7/3 F(n) - 1/3 F(n + 1) - 253/6 F(n) F(n + 1) 6 2 5 4 3 + 125/3 F(n) F(n + 1) - 275/3 F(n) F(n + 1) + 325/3 F(n) F(n + 1) 4 2 2 3 + 1/6 F(n) F(n + 1) + 79/6 F(n) F(n + 1) - 6 F(n) F(n + 1) 3 2 3 4 4 + 25/3 F(n) F(n + 1) - 25 F(n) F(n + 1) - 41/6 F(n) F(n + 1) 3 - 7/3 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, 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(2 n - 2 j) F(6 n - 6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 140 54 188887 6 2 5323 7 G(n) = ---- F(n) + ---- F(n + 1) - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 6061 6061 60610 6061 13627 2 6 22327 3 16372 7 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 12122 30305 30305 7721 2 2 19972 5 3 54 86 8 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- - ---- F(n) 30305 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, 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(2 n - 2 j) F(6 n - 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 2 22 7 16 3 G(n) = 26/3 F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 21 21 2 6 3 5 + 8/7 F(n) F(n + 1) - 16/3 F(n) F(n + 1) - 5/42 F(n) F(n + 1) 11 2 2 16 7 + -- F(n) - 5/42 + 5/42 F(n + 1) + -- F(n) F(n + 1) 84 21 2 2 8 - 24/7 F(n) F(n + 1) - 1/84 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, 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(2 n - 2 j) F(6 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2650 3 5 10950 7 25875 2 6 G(n) = ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 319 2300 4 4 63 3 63 281 2 - ---- F(n) F(n + 1) + --- F(n + 1) - --- + --- F(n) F(n + 1) 29 638 638 638 10695 3 59 2 7791 2 2 + ----- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 638 27 3 31 3 4 + -- F(n) F(n + 1) + --- F(n) + 1/638 F(n) 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, 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 ----- \ G(n) = ) F(j) F(2 n - 2 j) F(6 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 597 2 2 97 2 3 G(n) = -7/22 F(n) F(n + 1) - --- F(n) F(n + 1) - -- F(n) F(n + 1) 22 22 49 3 28 3 2 4 + -- F(n) F(n + 1) + -- F(n) F(n + 1) - 3/11 F(n) F(n + 1) 11 11 931 3 925 7 4375 2 6 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 11 22 4275 4 4 4 - ---- F(n) F(n + 1) - 5/22 F(n) - 2/11 F(n) - 9/22 F(n + 1) 22 225 3 5 + --- F(n) F(n + 1) + 9/22 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, 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(2 n - 2 j) F(6 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 3 3 G(n) = 23/6 + 993/2 F(n + 1) + 325/2 F(n) F(n + 1) - 629/3 F(n) F(n + 1) 8 3 5 - 500 F(n + 1) - 31 F(n) F(n + 1) - 925/6 F(n) F(n + 1) 2 6 7 - 25/3 F(n) F(n + 1) + 127/6 F(n) F(n + 1) + 3650/3 F(n) F(n + 1) 2 2 2 4 3 5 + 515/6 F(n) F(n + 1) - 175/6 F(n) F(n + 1) - 1050 F(n) F(n + 1) 2 2 6 - 67 F(n + 1) - 23/6 F(n) + 200/3 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, 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(2 n - 2 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 G(n) = -9/22 F(n + 1) + 9/22 F(n + 1) + 3/11 F(n) F(n + 1) 2 2 2 3 - 7/22 F(n) F(n + 1) - 6/11 F(n) + 1/22 F(n) F(n + 1) + 6/11 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, 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(2 n - 2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = 1/6 - 1/6 F(n + 1) + 1/6 F(n) F(n + 1) - 1/4 F(n) 2 2 3 4 + 7/6 F(n) F(n + 1) - 7/6 F(n) F(n + 1) + 1/12 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, 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(2 n - 2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 63 63 2 47 207 135 2 G(n) = ---- F(n + 1) + --- F(n + 1) + -- F(n) - --- F(n) F(n + 1) + --- F(n) 638 638 58 319 319 249 2 3 1641 3 2 489 4 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 638 638 319 787 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, 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(2 n - 2 j) F(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 4 G(n) = -5/6 F(n + 1) (-4 F(n + 1) + 4 F(n + 1) + F(n) - 9 F(n + 1) F(n) 2 3 3 2 - 3 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, 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(2 n - 2 j) F(6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 498 2 552 7 552 6 72087 7 G(n) = ----- F(n + 1) - ---- F(n) + ---- F(n + 1) - ----- F(n + 1) 6061 6061 6061 60610 996 1104 5 23431 4 3 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 6061 6061 46638 5 2 1365 6 2481 2 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 30305 6061 60610 33425 6 1104 5 71547 3 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n + 1) 12122 6061 60610 552 6 + ---- 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, 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 ----- \ 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) 3 2 3 G(n) = 9/22 - 9/22 F(n + 1) + 6/11 F(n) F(n + 1) + 5/22 F(n) F(n + 1) 19 2 2 2 19 3 3 - -- F(n) F(n + 1) + 3/2 F(n) F(n + 1) + -- F(n) - 3/22 F(n) F(n + 1) 11 22 14 4 - -- 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, 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(2 n - 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 2 G(n) = -1/12 (-2 F(n + 1) + F(n)) (4 F(n) - 11 F(n) F(n + 1) - 5 F(n) 2 2 2 + 11 F(n) F(n + 1) + F(n) F(n + 1) + 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, 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(2 n - 2 j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 589 2 391 2 574 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 319 3765 2 4 2065 5 295 3 3 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 638 319 319 1145 6 63 3 431 3 541 2 431 2 - ---- F(n + 1) + --- F(n + 1) - --- F(n) + --- F(n + 1) + --- F(n) 638 638 638 319 638 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, 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(2 n - 2 j) F(n + 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 5 G(n) = 1/30 F(n) (165 F(n) F(n + 1) - 67 F(n) + 67 F(n) + 4 F(n + 1) 5 3 2 + 26 F(n + 1) - 195 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, 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(2 n - 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = - 9/22 + 9/22 F(n + 1) - 1/22 F(n) - 4 F(n) F(n + 1) 69 2 3 3 31 3 2 4 + -- F(n) F(n + 1) + 4 F(n) F(n + 1) - -- F(n) F(n + 1) - F(n) 22 22 47 4 16 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, 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(2 n - 2 j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 2 2 3 G(n) = - 1/6 - 1/6 F(n) F(n + 1) - 2 F(n) F(n + 1) + 2 F(n) F(n + 1) 5 2 2 4 - 13/3 F(n) F(n + 1) + 1/6 F(n + 1) + 2/3 F(n) - 1/2 F(n) 29 2 4 3 3 71 4 2 + -- F(n) F(n + 1) - 4 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, 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(2 n - 2 j) F(2 n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2913 2 63 63 3 701 3 G(n) = F(n) - ---- F(n) F(n + 1) + --- - --- F(n + 1) - --- F(n) 638 638 638 638 1890 6 5710 2 5 215 4 3 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 638 3 2273 2 2 2 - F(n) F(n + 1) + ---- F(n) F(n + 1) + 5/2 F(n) F(n + 1) 638 3 9085 3 4 - 3 F(n) F(n + 1) + ---- 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, 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(2 n - 2 j) F(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 460 3 3 3 2 105 2 4 G(n) = --- F(n) F(n + 1) + 33 F(n) F(n + 1) + --- F(n) F(n + 1) 11 22 2 3 1135 5 4 - 13 F(n) F(n + 1) - ---- F(n) F(n + 1) - 24 F(n) F(n + 1) 22 465 6 25 25 2 5 + 9 F(n) F(n + 1) + --- F(n + 1) + -- F(n) - -- F(n) + 11 F(n + 1) 22 11 11 251 228 2 - --- 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, 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(2 n - 2 j) F(3 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 13 3 G(n) = -169/6 F(n) F(n + 1) + 1/6 F(n + 1) - -- F(n) - 1/6 F(n + 1) 12 13 5 335 6 715 2 5 + -- F(n) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 12 12 12 905 4 3 4 2 + --- F(n) F(n + 1) + 1/12 F(n) F(n + 1) + 23/3 F(n) F(n + 1) 12 2 3 53 3 2 3 4 - 19/6 F(n) F(n + 1) + -- F(n) F(n + 1) - 85/4 F(n) F(n + 1) 12 4 - 10/3 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, 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(2 n - 2 j) F(4 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 4 2 81 2 G(n) = 9/22 F(n + 1) + 35 F(n) F(n + 1) - 9/22 F(n + 1) - -- F(n) 22 471 2 5 81 3 4 2 + --- F(n) F(n + 1) - 20 F(n) F(n + 1) + -- F(n) - 65 F(n) F(n + 1) 11 22 226 955 6 369 2 + --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 11 22 22 2445 2 5 3 3 135 3 4 + ---- F(n) F(n + 1) + 30 F(n) F(n + 1) - --- F(n) F(n + 1) 22 22 970 4 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, 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 ----- \ 2 G(n) = ) F(j) F(2 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 3 G(n) = 1/2 - 1/2 F(n + 1) + 1/2 F(n) F(n + 1) + F(n) F(n + 1) 2 2 2 3 3 - 1/2 F(n) F(n + 1) - 1/2 F(n) F(n + 1) - F(n) + F(n) F(n + 1) 4 - 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, 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(2 n - j) F(3 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 G(n) = -1/11 F(n + 1) + 1/11 F(n + 1) + 7/11 F(n) - 1/11 F(n) F(n + 1) 13 2 2 3 3 2 - -- F(n) + 3/22 F(n) F(n + 1) + 3/2 F(n) F(n + 1) 22 17 4 5 - -- F(n) F(n + 1) - 1/22 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, 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(2 n - j) F(3 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 G(n) = 1/3 F(n + 1) - 1/3 F(n + 1) - 1/6 F(n) + 1/2 F(n) F(n + 1) 2 2 3 3 3 2 - 5/6 F(n) F(n + 1) + 2 F(n) F(n + 1) - 1/6 F(n) - 3 F(n) F(n + 1) 4 5 + 7/3 F(n) F(n + 1) - 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, 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 ----- \ 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) 3 2 2 G(n) = - 1/2 + 1/2 F(n + 1) + F(n) - 1/2 F(n) F(n + 1) - 3/2 F(n) F(n + 1) 2 3 3 3 2 4 + 3/2 F(n) F(n + 1) - 1/2 F(n) F(n + 1) + 4 F(n) F(n + 1) + F(n) 4 5 - 9/2 F(n) F(n + 1) - 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, 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(2 n - j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) G(n) = - 2 2 2 1/2 F(n) F(n + 1) (-F(n + 1) + F(n)) (F(n + 1) - 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, 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(2 n - j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 36 5 2 G(n) = F(n) F(n + 1) + -- F(n) F(n + 1) - 5/11 F(n) F(n + 1) 11 2 17 2 4 3 3 - 5/22 F(n) F(n + 1) - -- F(n) F(n + 1) + 9/11 F(n) F(n + 1) 11 41 4 2 19 2 2 3 - -- F(n) F(n + 1) - -- F(n) - 1/11 F(n + 1) - 3/22 F(n) 22 22 3 + 1/11 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, 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(2 n - j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 4 5 G(n) = 1/3 F(n + 1) + 3/2 F(n) - 1/3 - 1/6 F(n) - 1/3 F(n) F(n + 1) 2 2 3 5 - 11/6 F(n) F(n + 1) + 5/6 F(n) F(n + 1) - 7 F(n) F(n + 1) 2 4 3 3 4 2 + 7/3 F(n) F(n + 1) - 3 F(n) F(n + 1) + 23/3 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, 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(2 n - j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 2 4 3 3 G(n) = -5/2 F(n) F(n + 1) + 1/2 F(n) F(n + 1) + 4 F(n) F(n + 1) 4 4 2 5 + 3/2 F(n) F(n + 1) - 8 F(n) F(n + 1) + 6 F(n) F(n + 1) 4 5 2 - 1/2 F(n) F(n + 1) + F(n) F(n + 1) - 1/2 F(n + 1) + 1/2 F(n + 1) 2 5 - 1/2 F(n) - 3/2 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, 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(2 n - j) F(5 n - 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 200 2 295 2 2 10955 3 G(n) = --- F(n) F(n + 1) - --- F(n) + 5/638 F(n + 1) - ----- F(n + 1) 29 638 638 295 3 5475 7 1395 2 + --- F(n) + ---- F(n + 1) - 5/638 F(n) F(n + 1) - ---- F(n) F(n + 1) 638 319 638 750 2 5 13025 6 12275 3 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, 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(2 n - j) F(5 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 3 5 G(n) = 1/2 F(n) (-25 F(n + 1) + 65 F(n) F(n + 1) - 55 F(n) F(n + 1) 6 2 2 4 + 25 F(n + 1) - 2 F(n) + 7 F(n) F(n + 1) - 15 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, 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(2 n - j) F(5 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 190 2 3 16 3 G(n) = 1/11 + --- F(n) F(n + 1) - 1/11 F(n + 1) + -- F(n) 11 11 3 131 2 15 2 2 - 7/11 F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) F(n + 1) 22 22 10 3 225 3 4 4 - -- F(n) F(n + 1) + --- F(n) F(n + 1) - 6/11 F(n) 11 22 175 6 400 2 5 925 4 3 - --- 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, 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(2 n - j) F(5 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 4 7 G(n) = -7/3 F(n) + 79/6 F(n) F(n + 1) - 31/6 F(n) F(n + 1) - 325/3 F(n + 1) 2 3 2 5 3 2 - 29/6 F(n) F(n + 1) + 50/3 F(n) F(n + 1) + 23/3 F(n) F(n + 1) 3 5 2 + 326/3 F(n + 1) + 5/2 F(n + 1) - 253/6 F(n) F(n + 1) + 1/3 F(n) 6 3 4 - 17/6 F(n + 1) + 775/3 F(n) F(n + 1) - 725/3 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, 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 ----- \ G(n) = ) F(j) F(2 n - j) F(5 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 5 2 4 3 4 G(n) = -25/2 F(n) F(n + 1) - 7/2 F(n) F(n + 1) + 26 F(n) F(n + 1) 199 6 13 2 13 6 - --- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 10 10 10 5 2 6 + 1/2 F(n) F(n + 1) + 52/5 F(n) F(n + 1) - 1/2 F(n + 1) 5 7 3 6 3 3 - 5 F(n) F(n + 1) + 1/2 F(n + 1) + F(n) + 2 F(n) + 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, 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(2 n - j) F(6 n - 6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 4 2 6 19 6 G(n) = -4/7 F(n) F(n + 1) + 4/7 F(n) F(n + 1) + 1/42 F(n + 1) - -- F(n) 42 2 19 2 6 3 5 - 1/14 F(n + 1) + -- F(n) F(n + 1) - 1/42 F(n) F(n + 1) 21 5 3 113 3 19 3 3 - 16/3 F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) F(n + 1) 42 21 151 4 4 5 751 6 2 - --- F(n) F(n + 1) - 6/7 F(n) F(n + 1) + --- F(n) F(n + 1) 84 84 17 4 + -- F(n) + 1/21 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, 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 ----- \ G(n) = ) F(j) F(2 n - j) F(6 n - 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 10638 3 233 2 4446 2 2 G(n) = ----- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 319 3 85 3 1768 3 234 2300 8 + 5/638 F(n + 1) - --- F(n) + ---- F(n) F(n + 1) - --- + ---- F(n + 1) 319 319 319 29 50137 4 67 2 53250 3 5 - ----- F(n + 1) - --- F(n) F(n + 1) + ----- F(n) F(n + 1) 638 638 319 61550 7 575 2 6 - ----- 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, 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(2 n - j) F(6 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 7 6 G(n) = 1/2 F(n) (-130 F(n + 1) + 130 F(n + 1) - 305 F(n) F(n + 1) 3 4 2 2 3 + 300 F(n) F(n + 1) + 48 F(n) F(n + 1) - 15 F(n) F(n + 1) + 2 F(n) 2 5 - 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, 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(2 n - j) F(6 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 370 2 2 57 2 3 126 3 G(n) = - 1/11 - --- F(n) F(n + 1) + -- F(n) F(n + 1) + --- F(n) F(n + 1) 11 22 11 27 3 2 14 4 49 4 942 3 - -- F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) + --- F(n) F(n + 1) 11 11 22 11 925 7 4375 2 6 4275 4 4 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 22 22 29 4 225 3 5 - -- F(n) F(n + 1) + --- F(n) F(n + 1) + 1/11 F(n + 1) + 7/22 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, 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(2 n - j) F(6 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 3 2 G(n) = -31 F(n) F(n + 1) + 79/2 F(n) F(n + 1) + 23/6 - 5/6 F(n) 3 2 2 6 - 629/3 F(n) F(n + 1) - 16 F(n + 1) - 25/3 F(n) F(n + 1) 5 4 8 6 - 211/6 F(n) F(n + 1) + 993/2 F(n + 1) - 500 F(n + 1) + 47/3 F(n + 1) 7 2 2 + 25/6 F(n) F(n + 1) + 3650/3 F(n) F(n + 1) + 515/6 F(n) F(n + 1) 2 4 3 5 - 67/6 F(n) F(n + 1) - 1050 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, 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(2 n - j) F(6 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 2 3 G(n) = -90 F(n + 1) - 43 F(n) F(n + 1) + 20 F(n) F(n + 1) 2 6 3 7 + 25/2 F(n) F(n + 1) + 198 F(n) F(n + 1) - 1125 F(n) F(n + 1) 2 2 2 2 5 + 31/2 F(n) F(n + 1) - 72 F(n) F(n + 1) - 35/2 F(n) F(n + 1) 3 5 3 6 + 975 F(n) F(n + 1) - 7/2 F(n) + 445/2 F(n) F(n + 1) 3 4 8 3 - 365/2 F(n) F(n + 1) + 925/2 F(n + 1) + 179/2 F(n + 1) 4 - 921/2 F(n + 1) - 3/2 Proof: Both 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 ----- \ 2 G(n) = ) F(j) F(2 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 2 G(n) = -1/2 F(n + 1) + 1/2 F(n + 1) + F(n) F(n + 1) - 1/2 F(n) F(n + 1) 2 2 - 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, 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(2 n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = 1/3 - 1/3 F(n + 1) + 1/3 F(n) F(n + 1) - 1/2 F(n) 2 2 3 4 + 11/6 F(n) F(n + 1) - 17/6 F(n) F(n + 1) + 7/6 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, 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(2 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = -1/11 F(n + 1) + 1/11 F(n + 1) + F(n) - 5/11 F(n) F(n + 1) + 7/22 F(n) 2 3 17 3 2 18 4 73 5 + 1/22 F(n) F(n + 1) + -- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) 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, 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(2 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 2 4 3 3 G(n) = 193/2 F(n) F(n + 1) + 21/2 F(n) F(n + 1) - 191/2 F(n) F(n + 1) 2 2 6 - 29/2 F(n) F(n + 1) + 3 F(n) + 41 F(n + 1) - 41 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, 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 ----- \ G(n) = ) F(j) F(2 n - j) F(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 35 2 262425 6 G(n) = 5/638 F(n + 1) + --- F(n) + ------ F(n) F(n + 1) 638 638 227075 3 4 53925 7 107845 3 3225 3 - ------ F(n) F(n + 1) - ----- F(n + 1) + ------ F(n + 1) - ---- F(n) 638 319 638 638 30 555 2 1775 2 5 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 22 638 2110 2 - ---- 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, 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(2 n - j) F(6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 12547 2 2 1957 3 G(n) = 1/21 F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 84 42 2 74755 2 6 17065 4 4 - 1/14 F(n) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 84 21 2 337 4 5195 3 5195 7 - 1/21 F(n + 1) + --- F(n) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 42 14 14 1405 3 5 - ---- F(n) F(n + 1) + 1/21 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, 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(2 n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 3 G(n) = 1/2 - 1/2 F(n + 1) + F(n) F(n + 1) + 1/2 F(n) F(n + 1) 2 2 2 3 3 - 2 F(n) F(n + 1) + 3/2 F(n) F(n + 1) + 1/2 F(n) - 3/2 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, 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(2 n - j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 G(n) = 1/3 F(n + 1) - 1/3 F(n + 1) - 1/6 F(n) + 1/2 F(n) F(n + 1) 2 2 3 3 - 5/6 F(n) F(n + 1) + 8/3 F(n) F(n + 1) + 1/3 F(n) 3 2 4 5 - 11/2 F(n) F(n + 1) + 31/6 F(n) F(n + 1) - 13/6 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, 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(2 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 17 3 1047 5 919 3 3 G(n) = 1/11 F(n + 1) - -- F(n) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 22 11 11 859 2 83 2 2 25 2 + --- F(n + 1) + -- F(n) - 9/11 F(n) F(n + 1) + -- F(n) F(n + 1) 22 22 22 171 13 2 4 861 6 - --- F(n) F(n + 1) + -- F(n) F(n + 1) - --- 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, 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(2 n - j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 5 G(n) = -1/10 F(n) (305 F(n) F(n + 1) - 169 F(n) F(n + 1) 5 6 2 - 190 F(n) F(n + 1) + 83 F(n + 1) - 83 F(n + 1) + 4 F(n) F(n + 1) 6 + 50 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, 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(2 n - j) F(n + 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 43205 4 4 3 2612 4 115 3 G(n) = ----- F(n) F(n + 1) + 5/638 F(n + 1) - 5/638 + ---- F(n) - --- F(n) 58 319 638 61 2 8505 3 5 91187 2 2 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 638 638 141 2 110655 7 220953 3 + --- F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 638 319 638 29257 3 534165 2 6 - ----- 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, 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 ----- \ 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) 2 2 G(n) = - 1/2 + 1/2 F(n + 1) + 1/2 F(n) - 4 F(n) F(n + 1) 2 3 3 3 2 4 + 7/2 F(n) F(n + 1) + 4 F(n) F(n + 1) - 5/2 F(n) F(n + 1) - F(n) 4 5 + 1/2 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, 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(2 n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 5 2 4 G(n) = 50/3 F(n) F(n + 1) - 109/6 F(n) F(n + 1) + 1/2 F(n) F(n + 1) 3 3 3 3 + 4/3 F(n) F(n + 1) - 4/3 F(n) F(n + 1) + 2/3 F(n) F(n + 1) 5 2 2 2 4 - F(n) F(n + 1) - 5/3 F(n) F(n + 1) + 23/6 F(n) - 5/6 F(n) 4 6 - 1/3 F(n + 1) + 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, 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(2 n - j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 4720 6 7805 3 4 G(n) = F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 22 3835 7 12 4 866 2 3833 3 - ---- F(n + 1) + -- - F(n + 1) - --- F(n) F(n + 1) + ---- F(n + 1) 22 11 11 22 67 3 637 2 2 2 - -- F(n) + --- F(n) F(n + 1) + 7/2 F(n) F(n + 1) 11 22 395 2 5 3 - --- F(n) F(n + 1) - 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, 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(2 n - j) F(2 n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 6 8 7 G(n) = 8 - 145/2 F(n) F(n + 1) - 1755/2 F(n + 1) + 4305/2 F(n) F(n + 1) 3 5 2 2 3 - 1800 F(n) F(n + 1) + 331/2 F(n) F(n + 1) - 64 F(n) F(n + 1) 3 4 - 763/2 F(n) F(n + 1) + 1739/2 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, 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(2 n - j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 5 6 G(n) = -23/2 F(n + 1) + 17 F(n + 1) + 11 F(n + 1) - 33/2 F(n + 1) + 2 F(n) 2 2 3 5 + F(n) - 5 F(n) F(n + 1) - 13 F(n) F(n + 1) + 77/2 F(n) F(n + 1) 4 2 4 3 3 - 24 F(n) F(n + 1) + 19/2 F(n) F(n + 1) - 42 F(n) F(n + 1) 3 2 + 33 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, 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(2 n - j) F(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 3 4 7 G(n) = 615 F(n) F(n + 1) - 3355/6 F(n) F(n + 1) - 1535/6 F(n + 1) 2 5 3 3 - 623/6 F(n) F(n + 1) + 11/2 F(n + 1) + 1537/6 F(n + 1) - 37/6 F(n) 4 2 + 7/6 F(n) - 35/6 F(n + 1) - 12 F(n) F(n + 1) + 100/3 F(n) F(n + 1) 2 3 2 5 3 2 - 13/2 F(n) F(n + 1) + 65/3 F(n) F(n + 1) + 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, 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(2 n - j) F(3 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3657 2 2 2 3 G(n) = 37/2 F(n) F(n + 1) + ---- F(n) F(n + 1) + 19/2 F(n) F(n + 1) 22 3 2 8123 4 3750 3 - 27 F(n) F(n + 1) + ---- F(n + 1) - ---- F(n) F(n + 1) 11 11 20455 7 16210 3 5 41 100 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - -- F(n) + --- F(n + 1) 11 11 22 11 16465 8 217 1515 2 6 799 3 - ----- F(n + 1) + --- - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 22 22 11 11 5 - 9 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, 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 ----- \ 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 5 2 5 G(n) = -32 F(n) F(n + 1) - 20 F(n) F(n + 1) - 135/2 F(n) F(n + 1) 3 3 3 4 4 3 + 30 F(n) F(n + 1) - 45/2 F(n) F(n + 1) + 85 F(n) F(n + 1) 6 2 + 20 F(n) F(n + 1) + 65/2 F(n) F(n + 1) + 19/2 F(n) F(n + 1) 2 2 3 3 - 1/2 F(n + 1) - 7/2 F(n) + 1/2 F(n + 1) - 3/2 F(n) 2 4 4 2 + 35 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, 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 ----- \ 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) 2 4 3 3 3 G(n) = -19/6 F(n) F(n + 1) - 60 F(n) F(n + 1) + 14/3 F(n) F(n + 1) 4 2 2 2 5 - 71/6 F(n) F(n + 1) - 1/3 F(n + 1) - 2 F(n) + 1/3 F(n) F(n + 1) 7 2 2 2 6 + 3101/6 F(n) F(n + 1) + 403/2 F(n) F(n + 1) - 3686/3 F(n) F(n + 1) 3 5 4 4 5 - 81 F(n) F(n + 1) + 6967/6 F(n) F(n + 1) + 31/3 F(n) F(n + 1) 8 4 3 + 1/3 F(n + 1) + 10 F(n) - 3109/6 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, 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 ----- \ G(n) = ) F(j) F(2 n - j) F(5 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 4 199 8 8 G(n) = 11/2 F(n) - 87/5 F(n) + --- F(n) + 1/2 F(n + 1) 10 4 3 2 6 - 651/2 F(n) F(n + 1) + 125 F(n) F(n + 1) - 124 F(n) F(n + 1) 2 5 7 3 + 551/2 F(n) F(n + 1) - 1/2 F(n + 1) + 59/2 F(n) F(n + 1) 7 2 2 2 - 30 F(n) F(n + 1) - 73/2 F(n) F(n + 1) - 75/2 F(n) F(n + 1) 2 6 233 3 3 4 + 209/2 F(n) F(n + 1) + --- F(n) F(n + 1) + 74 F(n) F(n + 1) 10 863 3 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, 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(3 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 188 4 80 5 376 2 3 G(n) = --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 319 205 2 4 188 3 2 310 3 3 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 376 4 655 4 2 20 5 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) F(n + 1) 319 319 29 128 128 2 188 2 188 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, 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 ----- \ 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) 4 2 3 G(n) = -1/2 F(n + 1) (4 F(n) - 30 F(n) F(n + 1) - 6 F(n) F(n + 1) 3 2 5 + 32 F(n) F(n + 1) - 13 F(n + 1) + 13 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, 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 ----- \ 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) 61 75 2 4 3 3 G(n) = -- F(n) F(n + 1) - -- F(n) F(n + 1) + 4/11 F(n) + 1/11 F(n + 1) 22 22 185 6 2 27 2 + --- F(n + 1) + 1/22 F(n) F(n + 1) - -- F(n) F(n + 1) 22 22 230 3 3 210 5 2 2 + --- F(n) F(n + 1) - --- F(n) F(n + 1) - 17/2 F(n + 1) - 4/11 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, 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 ----- \ 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) 331 3 3 G(n) = --- F(n + 1) - 3/28 F(n) + 3/28 F(n) - 3/14 F(n + 1) 28 775 6 725 3 4 67 2 + --- F(n) F(n + 1) - --- F(n) F(n + 1) - -- F(n) F(n + 1) 28 28 14 325 7 17 2 25 2 5 - --- F(n + 1) + -- F(n) F(n + 1) + -- F(n) F(n + 1) 28 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, 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 ----- \ 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) 162 3 5347 3 488 6 1939 2 G(n) = --- F(n) - ---- F(n + 1) - --- F(n + 1) + ---- F(n) F(n + 1) 319 319 319 319 1220 5 162 2 13025 6 + ---- F(n) F(n + 1) - --- F(n) - ----- F(n) F(n + 1) 319 319 319 683 2 750 2 5 1220 3 3 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 12275 3 4 244 5 360 2 + ----- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n + 1) 319 319 319 5475 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, 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(3 n - 3 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 21 19 3 49 2 63 2 2 G(n) = - -- - -- F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 11 22 125 2 5 35 3 179 2 - --- F(n) F(n + 1) + -- F(n) F(n + 1) + --- F(n) F(n + 1) 22 11 11 927 3 6 2075 3 4 - --- F(n + 1) - 100 F(n) F(n + 1) + ---- F(n) F(n + 1) 22 22 925 7 23 4 21 3 + --- F(n + 1) + -- 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, 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(3 n - 3 j) F(5 n - 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 55 5 225 4 110 3 2 55 4 G(n) = --- F(n + 1) + ---- F(n) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 551 1102 551 551 7450 7 3255 2 2 55 2 3 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 551 551 551 110 4 55 4 1800 3 5 - --- F(n) F(n + 1) - --- F(n + 1) + ---- F(n) F(n + 1) 551 551 551 640 3 225 34425 4 4 + --- F(n) F(n + 1) - ---- F(n) - ----- F(n) F(n + 1) 551 1102 1102 17600 2 6 15755 3 + ----- F(n) F(n + 1) + ----- F(n) 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, 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(3 n - 3 j) F(5 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 75 3 5 255 2 2 G(n) = --- F(n) F(n + 1) + 3/14 F(n) F(n + 1) + --- F(n) F(n + 1) 14 28 3 2 4 2 - 20/7 F(n) F(n + 1) - 1/28 F(n) - 5/28 F(n) - 3/14 F(n + 1) 325 7 1525 2 6 325 3 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 14 28 14 4 4 + 375/7 F(n) F(n + 1) + 3/14 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, 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(3 n - 3 j) F(5 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1739 3 10058 3 508 2 G(n) = ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 319 4214 2 2 61550 7 53250 3 5 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 319 128 3 350 3 350 575 2 6 - --- F(n + 1) + --- F(n) - --- + --- F(n) F(n + 1) 319 319 319 319 300 2 2300 8 24822 4 + --- F(n) F(n + 1) + ---- 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, 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(3 n - 3 j) F(5 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 171 4 367 7 4815 6 2 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 44 220 88 10457 2 6 126 4 513 397 8 + ----- F(n) F(n + 1) + --- F(n + 1) - --- + --- F(n) - 7/44 F(n + 1) 440 11 88 88 503 8 6113 3 71 3 2 - --- F(n + 1) + ---- F(n) F(n + 1) + -- F(n) F(n + 1) 88 220 22 387 2 2 123 2 3 29 5 5 - --- F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) + 1/4 F(n + 1) 440 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, 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(3 n - 3 j) F(6 n - 6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 2 3 3 G(n) = 1/3 F(n) F(n + 1) (-35 F(n + 1) F(n) - 17 F(n + 1) F(n) 6 5 2 - 20 F(n + 1) F(n) + 2 F(n + 1) + 13 F(n + 1) F(n) + 4 F(n) 2 4 + 53 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, 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 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(6 n - 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 211 283 35215 8 G(n) = ---- F(n) - --- F(n) F(n + 1) - ----- F(n) F(n + 1) 551 551 551 85480 2 7 2355 3 6 151685 4 5 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 551 1102 1102 211 2 55 2 1246 4 32777 2 3 + --- F(n) + --- F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 551 551 19 1102 11121 3 2 623 4 55 9 + ----- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n + 1) 1102 551 551 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, 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(3 n - 3 j) F(6 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 3063 8 G(n) = 1/7 F(n) + 9/28 F(n) F(n + 1) + ---- F(n) F(n + 1) 28 7431 2 7 3 6 3303 4 5 - ---- F(n) F(n + 1) - 22/7 F(n) F(n + 1) + ---- F(n) F(n + 1) 28 14 3 1543 4 11 2 - 3/14 F(n + 1) - ---- F(n) F(n + 1) - -- F(n) F(n + 1) 14 28 1315 2 3 109 4 243 3 2 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 28 28 14 9 - 1/7 F(n) + 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, 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(3 n - 3 j) F(6 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 24975 8 668 3 384 4 G(n) = ------ F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n + 1) 29 319 319 2314 3 512 44597 4 1550 2 2 + ---- F(n) F(n + 1) - --- + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 319 23471 2 3 14116 3 2 111400 9 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n + 1) 319 319 319 14175 2 7 224700 3 6 107988 5 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n + 1) 319 319 319 512 3284 + --- 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, 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(3 n - 3 j) F(6 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 25 18827 4 5 8771 8 G(n) = -- F(n) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 22 22 8727 4 1785 2 3 25 2 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - -- F(n) 22 11 11 485 2 4 21227 2 7 1347 3 2 + --- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 22 22 22 135 3 3 321 3 6 214 4 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 11 22 11 127 795 4 2 9 + --- F(n) F(n + 1) - --- F(n) F(n + 1) - 1/11 F(n + 1) 11 22 2 120 5 + 1/11 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, 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 ----- \ 2 G(n) = ) F(j) F(3 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 G(n) = -1/11 F(n + 1) + 1/11 F(n + 1) - 3/11 F(n) F(n + 1) 2 2 2 3 + 9/11 F(n) F(n + 1) - 5/11 F(n) - 6/11 F(n) F(n + 1) + 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, 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(3 n - 3 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 128 128 3 216 2 148 3 G(n) = --- - --- F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 319 600 2 362 2 2 272 3 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) 319 319 319 3 400 4 + 2/319 F(n) F(n + 1) - --- 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, 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(3 n - 3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 2 G(n) = -1/28 (-2 F(n + 1) + F(n)) (9 F(n) - 31 F(n) F(n + 1) - 12 F(n) 2 2 2 + 31 F(n) F(n + 1) + 3 F(n) F(n + 1) + 3 - 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, 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(3 n - 3 j) F(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4965 2 4 3840 3 3 2585 4 2 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 1102 551 1102 55 2 375 2 55 3 375 3 - --- F(n + 1) + --- F(n) + --- F(n + 1) - --- F(n) 551 551 551 551 685 5 915 5 480 2 + --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 551 551 551 645 2 + --- 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, 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(3 n - 3 j) F(6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 4 2 G(n) = -2/3 F(n + 1) (-2 F(n) - 10 F(n) F(n + 1) - 32 F(n + 1) 6 5 3 3 + 32 F(n + 1) - 75 F(n) F(n + 1) + 76 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, 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(3 n - 3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 25 3 G(n) = 1/11 - 1/11 F(n + 1) - 9/22 F(n) F(n + 1) + -- F(n) F(n + 1) 22 19 2 2 2 3 - -- F(n) F(n + 1) - 8/11 F(n) F(n + 1) + 7/22 F(n) 22 21 3 4 + -- 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, 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(3 n - 3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 128 128 92 16 3 G(n) = - --- + --- F(n + 1) + --- F(n) + --- F(n) F(n + 1) 319 319 319 319 1301 2 2 992 2 3 1269 3 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 319 128 3 2 310 4 976 4 346 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, 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 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 113 2 19 2 131 13 3 G(n) = ---- F(n + 1) + -- F(n) - --- F(n) F(n + 1) - -- F(n) F(n + 1) 56 28 28 14 2 2 83 3 75 3 3 19 - 5/2 F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) - -- 28 14 28 125 6 475 2 4 1025 4 2 + --- F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 56 56 56 13 4 + -- 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, 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(3 n - 3 j) F(n + 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1113 4 1722 2 1223 1145 3 G(n) = ----- F(n + 1) - ---- F(n) F(n + 1) + ---- + ---- F(n) F(n + 1) 1102 551 1102 1102 1834 2 1918 2 2 10915 2 5 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 551 551 551 2771 3 4615 6 28005 3 4 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 551 1102 1102 4885 7 4995 3 1223 3 + ---- F(n + 1) - ---- F(n + 1) - ---- F(n) 1102 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, 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(3 n - 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 14 3 G(n) = - 1/11 + 1/11 F(n + 1) + -- F(n) - 8/11 F(n) F(n + 1) 11 29 2 2 2 3 21 3 - -- F(n) F(n + 1) + 3/2 F(n) F(n + 1) + -- F(n) F(n + 1) 22 22 3 2 15 4 4 5 + 4 F(n) F(n + 1) - -- F(n) - 9/2 F(n) F(n + 1) - 1/2 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, 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(3 n - 3 j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 128 128 2 710 2 710 5 G(n) = ---- F(n + 1) + --- F(n + 1) - --- F(n) + --- F(n) 319 319 319 319 72 4 36 5 2058 2 3 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 2148 2 4 2799 3 2 2416 3 3 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 2089 4 1438 4 2 2946 5 - ---- 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, 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(3 n - 3 j) F(2 n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 31 3 1815 2 5 G(n) = -3/14 F(n + 1) + 3/14 F(n + 1) - -- F(n) - ---- F(n) F(n + 1) 28 28 2355 4 3 6 45 2 3 + ---- F(n) F(n + 1) + 215/7 F(n) F(n + 1) - -- F(n) F(n + 1) 28 14 123 3 2 705 3 4 4 + --- F(n) F(n + 1) - --- F(n) F(n + 1) - 23/7 F(n) F(n + 1) 28 28 4 115 2 869 2 31 5 + 3/28 F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) 14 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, 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(3 n - 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 111 219 2 109 5 6 12 G(n) = ---- F(n + 1) - --- F(n + 1) + --- F(n + 1) + 20 F(n + 1) + -- F(n) 22 11 22 11 12 2 141 261 4 5 - -- F(n) + --- F(n) F(n + 1) - --- F(n) F(n + 1) - 45 F(n) F(n + 1) 11 22 22 97 2 3 2 4 136 3 2 - -- F(n) F(n + 1) - 15/2 F(n) F(n + 1) + --- F(n) F(n + 1) 22 11 3 3 + 50 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(3 n - 3 j) F(3 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 128 6 2 4 276 5 2 G(n) = ---- F(n + 1) - 30 F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 9314 5 3912 6 1106 6 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 128 7 701 2 1148 7 447 6 3700 + --- F(n + 1) - --- F(n) + ---- F(n) - --- F(n) - ---- F(n) F(n + 1) 319 319 319 319 319 3044 2 5 4084 5 740 2 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 830 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, 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(3 n - 3 j) F(4 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 43 6 185 4 2 3 3 G(n) = --- F(n + 1) - --- F(n) F(n + 1) + 6/11 F(n) F(n + 1) 55 22 14 7 41 7 41 6 3 38 2 + -- F(n + 1) + -- F(n) - -- F(n) - 9/55 F(n + 1) + -- F(n + 1) 55 22 22 55 31 57 2 130 3 4 + -- F(n) F(n + 1) + -- F(n) F(n + 1) + --- F(n) F(n + 1) 55 55 11 493 5 357 4 3 753 5 2 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 110 22 55 6 - 15/2 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, 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 ----- \ G(n) = ) F(j) F(3 n - 3 j) F(5 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 221 3 335 3 4 4 4 G(n) = 1/11 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - 925/2 F(n) F(n + 1) 11 22 4441 3 7 181 2 + ---- F(n) F(n + 1) - 200 F(n) F(n + 1) - --- F(n) F(n + 1) 22 11 812 2 2 2 6 65 3 67 4 - --- F(n) F(n + 1) + 475 F(n) F(n + 1) + -- F(n) - -- F(n) 11 22 22 3 5 2685 4 3 560 6 + 50 F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 22 11 2505 2 5 1095 2 3 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - 1/11 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, 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 ----- \ 2 G(n) = ) F(j) F(3 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 2 4 20 3 3 G(n) = -9/22 F(n + 1) - 3/22 F(n) + 5/11 F(n) F(n + 1) + -- F(n) F(n + 1) 11 85 4 2 19 2 2 25 5 - -- F(n) F(n + 1) - -- F(n) + 9/22 F(n + 1) + -- F(n) F(n + 1) 22 22 11 2 2 + 1/22 F(n) F(n + 1) + 3/11 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, 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 ----- \ 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) 4 2 2 2 4 G(n) = 1/3 F(n) - 1/3 + 1/3 F(n + 1) + F(n) + 7/3 F(n) F(n + 1) 3 3 4 2 5 - 3 F(n) F(n + 1) + 23/3 F(n) F(n + 1) - 1/3 F(n) F(n + 1) 2 2 3 5 - 7/3 F(n) F(n + 1) + 4/3 F(n) F(n + 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, 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(3 n - 2 j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 750 3 4 315 4 2 2965 4 3 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 638 63 5 162 2 162 3 1465 2 5 + --- F(n) F(n + 1) - --- F(n) + --- F(n) + ---- F(n) F(n + 1) 319 319 319 638 173 2 2012 5 2 138 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 319 1047 2 315 2 4 63 7 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n + 1) 638 638 638 63 6 + --- 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, 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(3 n - 2 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 400 2 5 925 4 3 175 6 G(n) = --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 11 22 11 225 3 4 3 109 2 + --- F(n) F(n + 1) - 7/11 F(n) F(n + 1) - --- F(n) F(n + 1) 22 22 2 2 347 2 3 - 7/22 F(n) F(n + 1) + --- F(n) F(n + 1) - 9/22 + 1/11 F(n) F(n + 1) 22 3 21 3 4 + 9/22 F(n + 1) + -- F(n) + 5/11 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, 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(3 n - 2 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 4 3 6 G(n) = -16/3 F(n) F(n + 1) + 61/3 F(n) F(n + 1) + 17/2 F(n) F(n + 1) 6 17 5 127 3 19 5 - 19/2 F(n) F(n + 1) + -- F(n) - --- F(n + 1) - -- F(n + 1) 15 30 15 7 4 2 137 7 - 11/6 F(n) + 4 F(n) F(n + 1) - 2/15 F(n) F(n + 1) + --- F(n + 1) 30 13 14 238 5 2 - -- F(n) + -- F(n + 1) - --- F(n) F(n + 1) 10 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, 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 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(5 n - 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 3 6 55 2 G(n) = -5/21 F(n) F(n + 1) - 10/3 F(n) F(n + 1) + 5/42 F(n) - -- F(n) 84 2 6 75 5 3 695 3 15 - 175/4 F(n) F(n + 1) - -- F(n) F(n + 1) - --- F(n) F(n + 1) + -- 14 42 28 6 125 2 2 7 + 5/42 F(n + 1) + --- F(n) F(n + 1) + 125/7 F(n) F(n + 1) 14 5 4 4 - 5/42 F(n) F(n + 1) + 5/21 F(n) F(n + 1) + 300/7 F(n) F(n + 1) 55 4 - -- 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, 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 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(5 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2650 3 5 63 3 63 19 2 G(n) = ---- F(n) F(n + 1) + --- F(n + 1) - --- - --- F(n) F(n + 1) 319 638 638 319 87 3 21709 3 189 2 + -- F(n) F(n + 1) + ----- F(n) F(n + 1) - --- F(n) F(n + 1) 22 638 319 7791 2 2 159 4 257 3 2300 4 4 - ---- F(n) F(n + 1) - --- F(n) - --- F(n) - ---- F(n) F(n + 1) 638 319 638 29 10950 7 25875 2 6 - ----- 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, 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 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(5 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 50 2 6 69 4 356 2 2 G(n) = -- F(n) F(n + 1) + -- F(n) F(n + 1) - --- F(n) F(n + 1) 11 22 11 2 3 43 3 2 882 3 + 7/22 F(n) F(n + 1) - -- F(n) F(n + 1) + --- F(n) F(n + 1) 11 11 5200 7 4500 3 5 29 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - 2/11 F(n) + -- F(n + 1) 11 11 22 4275 8 19 5 20 2113 4 131 3 + ---- F(n + 1) - -- F(n + 1) - -- - ---- F(n + 1) + --- F(n) F(n + 1) 22 11 11 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, 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 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(5 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 2 G(n) = 5/3 F(n) F(n + 1) + 3650/3 F(n) F(n + 1) - 11 F(n + 1) 2 2 2 4 3 5 + 250/3 F(n) F(n + 1) - 41/3 F(n) F(n + 1) - 1050 F(n) F(n + 1) 8 6 2 6 - 500 F(n + 1) + 17/6 + 32/3 F(n + 1) - 25/3 F(n) F(n + 1) 5 4 3 - 68/3 F(n) F(n + 1) + 995/2 F(n + 1) - 55/2 F(n) F(n + 1) 3 3 3 2 + 32 F(n) F(n + 1) - 635/3 F(n) F(n + 1) + 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, 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 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(6 n - 6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2638 54 2 1208 4 388467 8 G(n) = ---- F(n) + ---- F(n + 1) + ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 6061 6061 19 6061 939729 2 7 28109 3 6 1667433 4 5 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ------- F(n) F(n + 1) 6061 12122 12122 54 9 2638 2 358 28369 2 3 - ---- F(n + 1) - ---- F(n) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 6061 6061 1102 119025 3 2 20077 4 + ------ 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, 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 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(6 n - 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 17 9 80 3 6 G(n) = -5/28 F(n) F(n + 1) - -- F(n) - 5/42 F(n + 1) - -- F(n) F(n + 1) 21 21 9895 4 5 9245 8 22265 2 7 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 42 84 84 229 3 2 509 4 4 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - 761/7 F(n) F(n + 1) 14 84 47 2 1231 2 3 3 3 - -- F(n) F(n + 1) + ---- F(n) F(n + 1) - 4/21 F(n) + 5/42 F(n + 1) 84 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, 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 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(6 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 63 9 63 2 6 37 3 G(n) = ---- F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 638 638 319 63 3 5 63 4 4 67 4 705 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) + --- F(n) 319 638 638 638 167 3 6 2646 4 222863 4 5 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 29 319 638 63 7 51862 8 481 2 2 - --- F(n) F(n + 1) - ----- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 638 20825 2 3 251213 2 7 16053 3 2 - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 319 638 638 51484 4 63 8 305 3 + ----- F(n) F(n + 1) + --- 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, 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 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(6 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1527 882 123 9 1526 9 G(n) = ----- F(n) + --- F(n + 1) - --- F(n + 1) + ---- F(n) 550 275 550 275 25851 5 4 27 2 167 5 + ----- F(n) F(n + 1) + --- F(n + 1) - --- F(n) F(n + 1) 550 110 55 2 4 2538 4 5 + 9/22 F(n) F(n + 1) + ---- F(n) F(n + 1) - 1/11 F(n) F(n + 1) 275 19297 6 3 4149 7 2 678 8 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 275 275 275 57 708 5 36 6 6 + -- F(n) F(n + 1) - --- F(n + 1) - -- F(n + 1) + 5/22 F(n) 55 275 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, 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 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(6 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 2 5 3 6 G(n) = -55/3 F(n) F(n + 1) + 95/3 F(n) F(n + 1) - 98/3 F(n) F(n + 1) 4 3 3 9 3 - 190/3 F(n) F(n + 1) - 1/3 F(n) + 1/3 F(n + 1) - 1/3 F(n + 1) 4 8 2 - 6127/6 F(n) F(n + 1) + 3073/3 F(n) F(n + 1) - 17/6 F(n) F(n + 1) 2 3 2 7 3 2 + 1270/3 F(n) F(n + 1) - 7426/3 F(n) F(n + 1) - 157 F(n) F(n + 1) 3 4 4 4 5 + 30 F(n) F(n + 1) + 283/6 F(n) F(n + 1) + 6601/3 F(n) F(n + 1) 2 - 14/3 F(n) + 113/6 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, 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 ----- \ 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) 3 2 3 G(n) = 9/22 - 9/22 F(n + 1) + 9/22 F(n) F(n + 1) + 4/11 F(n) F(n + 1) 18 2 19 2 2 15 3 16 3 - -- F(n) F(n + 1) + -- F(n) F(n + 1) + -- F(n) - -- F(n) F(n + 1) 11 11 22 11 4 - 1/11 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, 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 ----- \ 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) 3 2 G(n) = 1/6 F(n + 1) - 1/6 F(n + 1) - 1/12 F(n) + 1/4 F(n) F(n + 1) 2 2 3 3 - 3/4 F(n) F(n + 1) + 5/2 F(n) F(n + 1) + 1/3 F(n) 3 2 4 5 - 21/4 F(n) F(n + 1) + 21/4 F(n) F(n + 1) - 9/4 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, 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 ----- \ 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) 1194 1151 5 57 4 2 G(n) = ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + -- F(n) F(n + 1) 1595 319 58 621 2 405 2 2391 6 5139 6 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) - ---- F(n + 1) 638 319 638 3190 2412 2 63 3 477 3 13774 5 + ---- F(n + 1) + --- F(n + 1) - --- F(n) - ----- F(n) F(n + 1) 1595 638 638 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, 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 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 3 6 G(n) = 5/3 F(n) (-3 F(n) + 118 F(n) F(n + 1) + 49 F(n + 1) 5 2 4 2 - 113 F(n) F(n + 1) - 18 F(n) F(n + 1) - 49 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, 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 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1656 2 1050 3 54 3 108215 G(n) = ---- F(n) F(n + 1) - ---- F(n) + ---- F(n + 1) - ------ 6061 6061 6061 36366 6433651 7 2200833 6 2 92947 4 - ------- F(n) F(n + 1) + ------- F(n) F(n + 1) - ----- F(n + 1) 181830 30305 6061 1494 2 409293 3 397463 2 2 - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 6061 60610 16530 317195 7 405443 8 665573 8 - ------ F(n) F(n + 1) + ------ F(n) + ------ F(n + 1) 18183 36366 36366 Proof: Both 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 ----- \ 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) 3 G(n) = - 9/22 + 9/22 F(n + 1) + 1/11 F(n) - 3/22 F(n) F(n + 1) 42 2 2 39 2 3 45 3 - -- F(n) F(n + 1) + -- F(n) F(n + 1) + -- F(n) F(n + 1) 11 11 11 32 3 2 23 4 4 5 - -- F(n) F(n + 1) - -- F(n) + 9/11 F(n) F(n + 1) - 7/11 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, 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 ----- \ 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) 2 2 6 4 G(n) = 161/3 F(n + 1) + 15/4 F(n) - 107/2 F(n + 1) + 7/12 F(n + 1) - 3/4 5 3 3 221 + 501/4 F(n) F(n + 1) - 507/4 F(n) F(n + 1) - --- F(n) F(n + 1) 12 3 2 2 2 4 - 7/6 F(n) F(n + 1) - 5/2 F(n) F(n + 1) + 67/4 F(n) F(n + 1) 37 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, 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(3 n - 2 j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 49591 2 9995 4 3 2295 6 G(n) = ------ F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 58 29 5480 2 5 63 3 3877 3 312 4 63 - ---- F(n) F(n + 1) - --- F(n + 1) - ---- F(n) + --- F(n) + --- 29 638 638 319 638 335 3 18239 2 1641 2 2 - --- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 638 638 948 3 395 3 4 - --- F(n) F(n + 1) - --- F(n) F(n + 1) 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, 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(3 n - 2 j) F(n + 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 3 4 G(n) = 1/6 F(n) (951 F(n) F(n + 1) - 289 F(n) F(n + 1) + 5290 F(n) F(n + 1) 3 7 6 - 2400 F(n + 1) + 2400 F(n + 1) - 5740 F(n) F(n + 1) 2 5 3 - 260 F(n) F(n + 1) + 48 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, 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(3 n - 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 5 69 2 3 G(n) = 3/22 F(n) F(n + 1) - 2/11 F(n) F(n + 1) - -- F(n) F(n + 1) 11 53 2 4 195 3 2 54 3 3 + -- F(n) F(n + 1) + --- F(n) F(n + 1) - -- F(n) F(n + 1) 11 22 11 74 4 64 4 2 50 5 - -- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 11 11 11 2 19 2 47 5 - 9/22 F(n + 1) + 9/22 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, 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(3 n - 2 j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 2 5 4 3 G(n) = 105 F(n) F(n + 1) - 955/4 F(n) F(n + 1) + 1035/4 F(n) F(n + 1) 73 3 3 2 4 - -- F(n) + 1/6 F(n + 1) - 421/4 F(n) F(n + 1) + 1/12 F(n) F(n + 1) 12 2 2 3 53 3 2 + 67/2 F(n) F(n + 1) - 19/6 F(n) F(n + 1) + -- F(n) F(n + 1) 12 3 4 4 13 5 - 185/4 F(n) F(n + 1) - 10/3 F(n) F(n + 1) + -- F(n) - 1/6 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, 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(3 n - 2 j) F(2 n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 3134 42985 8 4 G(n) = -9 F(n + 1) + ---- - ----- F(n + 1) + 37/2 F(n) F(n + 1) 319 58 4759 2 2 2 3 3 2 + ---- F(n) F(n + 1) + 19/2 F(n) F(n + 1) - 27 F(n) F(n + 1) 29 107430 3 233252 4 582 5805 - ------ F(n) F(n + 1) + ------ F(n + 1) - --- F(n) + ---- F(n + 1) 319 319 319 638 53370 7 42390 3 5 3860 2 6 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 29 29 29 22976 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, 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 ----- \ G(n) = ) F(j) F(3 n - 2 j) F(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 320 2 31 3 79 2 G(n) = 9/22 F(n + 1) - --- F(n) F(n + 1) - -- F(n) - -- F(n) 11 22 22 270 3 4 315 6 3 3 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + 30 F(n) F(n + 1) 11 11 635 2 5 870 4 3 449 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 11 11 22 181 2 4 2 2 4 + --- F(n) F(n + 1) - 65 F(n) F(n + 1) + 35 F(n) F(n + 1) 22 5 2 - 20 F(n) F(n + 1) - 9/22 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, 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(3 n - 2 j) F(3 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 4 2 3 5 G(n) = 35/2 F(n) F(n + 1) + 29/3 F(n) - 11/6 F(n) - 155/2 F(n) F(n + 1) 5 2 6 - 10 F(n) F(n + 1) + 61/6 F(n) F(n + 1) - 1245 F(n) F(n + 1) 2447 2 2 7 4 2 + ---- F(n) F(n + 1) + 2095/4 F(n) F(n + 1) - 65/2 F(n) F(n + 1) 12 3 3 3 4 4 + 15 F(n) F(n + 1) - 182/3 F(n) F(n + 1) + 4685/4 F(n) F(n + 1) + 1/6 2 3 - 1/6 F(n + 1) - 2095/4 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, 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(3 n - 2 j) F(4 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 1719 3 1720 7 G(n) = -9/22 F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 11 811 2 1229 2 2 3930 2 6 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 22 22 11 171 3 3 4 4135 4 4 - --- F(n) F(n + 1) + 75 F(n) F(n + 1) + ---- F(n) F(n + 1) 11 11 6 2 5 41 4 - 125 F(n) F(n + 1) + 275 F(n) F(n + 1) + -- F(n) 22 2765 2 610 3 5 4 3 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) + 9/22 - 325 F(n) F(n + 1) 22 11 63 3 + -- 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, 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 ----- \ 2 G(n) = ) F(j) F(3 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 5 6 G(n) = F(n + 1) - 39/2 F(n + 1) - 3/2 F(n + 1) + 20 F(n + 1) + 1/2 F(n) 2 4 5 - 3/2 F(n) + 6 F(n) F(n + 1) + 2 F(n) F(n + 1) - 45 F(n) F(n + 1) 2 3 2 4 3 2 + F(n) F(n + 1) - 15/2 F(n) F(n + 1) - 11/2 F(n) F(n + 1) 3 3 + 50 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, 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(3 n - j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 G(n) = 5/2 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, 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(3 n - j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 3 3 G(n) = -9/11 F(n) F(n + 1) - 9/22 F(n) F(n + 1) - 3/22 F(n) F(n + 1) + 1/11 3 6 35 3 4 - 1/11 F(n + 1) - 9/44 F(n) F(n + 1) + -- F(n) F(n + 1) 22 95 5 2 117 6 47 2 4 + -- F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) F(n + 1) + 5/11 F(n) 11 11 44 145 3 125 7 + --- F(n) - --- F(n) 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, 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(3 n - j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 4 3 3 4 G(n) = -25 F(n) F(n + 1) + 1/3 F(n + 1) - 11/6 F(n) - 4/3 F(n) F(n + 1) 5 4 3 6 + 5/6 F(n) - 1/3 F(n + 1) + 325/3 F(n) F(n + 1) + 125/3 F(n) F(n + 1) 2 5 4 2 - 275/3 F(n) F(n + 1) + 1/6 F(n) F(n + 1) + 73/6 F(n) F(n + 1) 2 3 3 2 2 - 3 F(n) F(n + 1) + 11/6 F(n) F(n + 1) - 253/6 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, 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 ----- \ 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) 2 5 2 7 G(n) = 9 F(n + 1) + 41/2 F(n) F(n + 1) - 17/2 F(n) F(n + 1) - 25/2 F(n + 1) 7 6 6 2 + 2 F(n) - 19/2 F(n + 1) + 69/2 F(n) F(n + 1) + 11 F(n) F(n + 1) 3 3 2 4 2 5 - 26 F(n) F(n + 1) + 7/2 F(n) F(n + 1) - 33/2 F(n) F(n + 1) 6 3 - 37/2 F(n) F(n + 1) - 2 F(n) F(n + 1) + 13 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, 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(3 n - j) F(5 n - 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1455 4 14675 8 10 2 G(n) = ---- F(n + 1) - ----- F(n + 1) + --- F(n) F(n + 1) 319 6699 319 144775 7 161795 7 265 3 13150 8 + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) + --- F(n) + ----- F(n) 13398 13398 638 6699 3 24180 3 305 2 + 5/638 F(n + 1) - ----- F(n) F(n + 1) - --- F(n) F(n + 1) 2233 319 13025 2 2 27990 2 6 31865 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- 13398 2233 13398 Proof: Both 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(3 n - j) F(5 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 7 2 2 G(n) = -315 F(n) F(n + 1) + 365 F(n) F(n + 1) + 1 + 24 F(n) F(n + 1) 3 2 6 3 - 63 F(n) F(n + 1) - 5/2 F(n) F(n + 1) - 17/2 F(n) F(n + 1) 8 4 - 150 F(n + 1) + 149 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, 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(3 n - j) F(5 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 23 7 21 3 5 93 7 G(n) = -- F(n) F(n + 1) + -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 22 11 257 6 2 5 3 136 4 4 - --- F(n) F(n + 1) + 39/2 F(n) F(n + 1) - --- F(n) F(n + 1) 22 11 4 39 2 2 2 3 - 1/2 F(n) F(n + 1) - 1/11 - -- F(n) F(n + 1) - 1/22 F(n) F(n + 1) 11 14 3 2 4 5 - -- F(n) F(n + 1) + 4/11 F(n) F(n + 1) + 1/11 F(n + 1) - 2/11 F(n) 11 4 - 8/11 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, 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(3 n - j) F(5 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 2 4 2 G(n) = - 4/3 + 175/3 F(n) F(n + 1) - 29/3 F(n) F(n + 1) 2 6 5 3 34 5 + 25/2 F(n) F(n + 1) - 185/3 F(n) F(n + 1) + -- F(n) F(n + 1) 15 26 8 6 8 + -- F(n) F(n + 1) - 5/3 F(n + 1) - 5/6 F(n) + 25/6 F(n) 15 6 4 3 3 - 7/10 F(n + 1) + 10/3 F(n + 1) + 4 F(n) F(n + 1) 3 11 2 - 65/6 F(n) F(n + 1) + -- 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, 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(3 n - j) F(5 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 3 4 6 G(n) = -50 F(n + 1) - 235/2 F(n) F(n + 1) + 235/2 F(n) F(n + 1) 4 3 2 5 - 917/2 F(n + 1) + 1/2 F(n) - 7/2 + 15/2 F(n) F(n + 1) 3 5 3 7 + 975 F(n) F(n + 1) + 194 F(n) F(n + 1) - 1125 F(n) F(n + 1) 2 2 2 2 + 7/2 F(n) F(n + 1) - 77 F(n) F(n + 1) - 18 F(n) F(n + 1) 2 6 3 8 + 25/2 F(n) F(n + 1) + 27 F(n) F(n + 1) + 925/2 F(n + 1) 3 + 99/2 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, 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 ----- \ G(n) = ) F(j) F(3 n - j) F(6 n - 6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3231 5 825 2 7 G(n) = 1/14 F(n) F(n + 1) + ---- F(n + 1) - --- F(n) F(n + 1) 14 28 563 3 2 3 6 10 3 - --- F(n) F(n + 1) - 475 F(n) F(n + 1) + -- F(n) 21 21 8 8359 4 41 2 + 2325/4 F(n) F(n + 1) - ---- F(n) F(n + 1) - -- F(n) F(n + 1) 84 42 4237 2 3 9 10 209 + ---- F(n) F(n + 1) - 1650/7 F(n + 1) - -- F(n) + --- F(n + 1) 84 21 42 3 - 1/21 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, 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(3 n - j) F(6 n - 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 9 109 4 27 59 5 G(n) = -5/638 F(n + 1) + --- F(n) + -- F(n) + 5/638 + --- F(n) 319 58 319 15761 3 2 6631 3 6 2209 4 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 319 319 294 5 4 267 6 2 21697 6 3 + --- F(n) F(n + 1) - --- F(n) F(n + 1) - ----- F(n) F(n + 1) 11 319 319 81 2 7 267 7 1335 3 5 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 638 638 28269 4 5 1335 5 3 235 3 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 638 638 638 78 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, 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(3 n - j) F(6 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 8 G(n) = -F(n) + 660 F(n + 1) + 2027/2 F(n) F(n + 1) + 3255/2 F(n) F(n + 1) 2 7 3 6 3 2 - 165/2 F(n) F(n + 1) - 1330 F(n) F(n + 1) - 1367 F(n) F(n + 1) 4 2 3 9 - 647 F(n) F(n + 1) + 1573/2 F(n) F(n + 1) - 660 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, 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(3 n - j) F(6 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 28881 4 26359 22411 109 2 G(n) = ------ F(n) F(n + 1) - ----- F(n + 1) + ----- F(n) + --- F(n + 1) 88 176 264 22 2 119 5 148 3 3 107 6 + 8/11 F(n) + --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n + 1) 11 11 22 22075 9 26343 5 15152 3 2 43 - ----- F(n) + ----- F(n + 1) + ----- F(n) F(n + 1) - -- F(n) F(n + 1) 264 176 33 22 14425 8 106475 8 1963 2 3 - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 264 528 88 31 2 4 55025 2 7 + -- F(n) F(n + 1) + ----- F(n) F(n + 1) 22 528 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, 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(3 n - j) F(6 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 4 3 2 5 G(n) = 5/3 F(n) - 14/3 F(n) - 265/3 F(n) F(n + 1) + 215/3 F(n) F(n + 1) 3 4 4 3 + 20 F(n) F(n + 1) - 6151/6 F(n) F(n + 1) - 1/3 F(n + 1) 3 6 4 4 5 - 100/3 F(n) F(n + 1) + 75/2 F(n) F(n + 1) + 2200 F(n) F(n + 1) 2 3 2 7 3 2 + 433 F(n) F(n + 1) - 2475 F(n) F(n + 1) - 153 F(n) F(n + 1) 6 8 2 - 100/3 F(n) F(n + 1) + 1025 F(n) F(n + 1) - 59/6 F(n) F(n + 1) 2 + 203/6 F(n) F(n + 1) + 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, 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(3 n - j) F(6 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 9 G(n) = - 1/2 - 3979/2 F(n + 1) + 11/2 F(n) - 95/2 F(n + 1) + 4075/2 F(n + 1) 4 3 8 + 475/2 F(n + 1) - 7/2 F(n) F(n + 1) - 475/2 F(n + 1) 4 8 2 2 + 1701/2 F(n) F(n + 1) - 5025 F(n) F(n + 1) + 63/2 F(n) F(n + 1) 2 3 3 2 3 6 - 434 F(n) F(n + 1) + 239 F(n) F(n + 1) + 8225/2 F(n) F(n + 1) 3 7 2 7 - 203/2 F(n) F(n + 1) + 575 F(n) F(n + 1) + 525/2 F(n) F(n + 1) 3 5 - 1025/2 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, 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 ----- \ 2 G(n) = ) F(j) F(3 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 2 G(n) = 1/2 - 1/2 F(n + 1) + 3/2 F(n) F(n + 1) - 3 F(n) F(n + 1) 2 2 3 3 4 + 3 F(n) F(n + 1) + 3/2 F(n) - 2 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, 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(3 n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 2 G(n) = -1/6 (-2 F(n + 1) + F(n)) (9 F(n) - 15 F(n) F(n + 1) - 4 F(n) 2 2 2 + 9 F(n) F(n + 1) + F(n) F(n + 1) + 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, 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(3 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 340 2 28 2 5 3 3 G(n) = --- F(n + 1) + -- F(n) + 76 F(n) F(n + 1) - 69 F(n) F(n + 1) 11 11 281 4 2 3 3 - --- F(n) F(n + 1) + 7/2 F(n) F(n + 1) + 1/11 F(n + 1) - 6/11 F(n) 22 15 2 21 2 6 - -- F(n) F(n + 1) + -- F(n) F(n + 1) - 31 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, 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 ----- \ G(n) = ) F(j) F(3 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 5 2 3 4 G(n) = F(n) (2 F(n + 1) + 4 F(n) F(n + 1) + 33 F(n) F(n + 1) 3 2 5 - 36 F(n) F(n + 1) + 6 F(n) - 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, 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(3 n - j) F(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 45 7 20035 25 3 345 6 G(n) = ---- F(n + 1) + ----- + --- F(n + 1) + --- F(n) F(n + 1) 638 1276 319 638 45 3 4 15 2 5 185 3 60 7 - --- F(n) F(n + 1) + -- F(n) F(n + 1) - --- F(n) + --- F(n) 319 29 638 319 13525 8 10185 8 40415 4 1880 3 - ----- F(n) + ----- F(n + 1) - ----- F(n + 1) - ---- F(n) F(n + 1) 1276 638 1276 29 81865 2 6 14565 7 46125 6 2 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 1276 638 319 5710 5 3 285 4 3 + ---- 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, 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(3 n - j) F(6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 17 3 3 13 2 5 G(n) = --- F(n) - 1/12 F(n + 1) - -- F(n) F(n + 1) - 1/84 F(n + 1) 210 70 4 3 1647 3 6 3133 7 2 + 5/28 F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 14 28 563 8 4297 2 3 2 7 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) + 374/7 F(n) F(n + 1) 12 84 41 3 2 3 4 7 + -- F(n) F(n + 1) + 5/84 F(n + 1) + 1/14 F(n) F(n + 1) + 1/70 F(n) 28 7 170 9 7855 6 3 29 2 5 + 1/28 F(n + 1) - --- F(n) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 21 42 140 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, 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(3 n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 2 G(n) = - 1/2 + 1/2 F(n + 1) + 1/2 F(n) F(n + 1) - 5 F(n) F(n + 1) 2 3 3 3 2 4 + 4 F(n) F(n + 1) + 4 F(n) F(n + 1) - 4 F(n) F(n + 1) - 1/2 F(n) 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, 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 ----- \ 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) 4 5 2 5 G(n) = - 1/3 - 1/6 F(n) - 95/6 F(n) F(n + 1) + 5/2 F(n) - 1/3 F(n) F(n + 1) 2 2 3 2 - 7/3 F(n) F(n + 1) + 7/3 F(n) F(n + 1) + 1/3 F(n + 1) 2 4 3 3 4 2 + 13/6 F(n) F(n + 1) - 17/3 F(n) F(n + 1) + 52/3 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, 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(3 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 13 279 3 12 4 14 2 G(n) = -- - --- F(n + 1) - -- F(n + 1) + -- F(n) F(n + 1) 11 44 11 11 423 2 134 6 46 7 37 2 2 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - -- F(n) + -- F(n) F(n + 1) 44 11 11 11 19 2 5 115 3 29 3 + -- F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) F(n + 1) 22 22 22 1073 6 7 + ---- F(n) F(n + 1) + 25/4 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, 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(3 n - j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 5 G(n) = -5/2 F(n) (-2 F(n) - 120 F(n) F(n + 1) + 15 F(n) F(n + 1) 3 3 6 2 4 2 + 146 F(n) F(n + 1) + 54 F(n + 1) - 38 F(n) F(n + 1) - 55 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, 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(3 n - j) F(n + 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1844459 5 65325 9 8 G(n) = ------- F(n + 1) - ----- F(n + 1) + 14725/2 F(n) F(n + 1) 638 22 216 2 2 218908 2 3 126540 3 2 + --- F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 319 319 319 401417 4 92 3 11075 2 7 - ------ F(n) F(n + 1) + --- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 22 64700 3 6 373 3 163 4 84 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n + 1) + --- 11 319 638 319 2636 49961 - ---- 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, 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(3 n - j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 5 G(n) = 5/2 F(n) - 1/2 F(n) + 1/2 F(n + 1) - 1/2 F(n + 1) - F(n + 1) 6 2 3 2 4 + F(n + 1) - 2 F(n) F(n + 1) - F(n) F(n + 1) + F(n) F(n + 1) 3 2 3 3 4 + 9 F(n) F(n + 1) - 9 F(n) F(n + 1) - 12 F(n) F(n + 1) 4 2 + 12 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, 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(3 n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 27 2 3 4 5 5 G(n) = --- F(n) F(n + 1) - 53/3 F(n) F(n + 1) - 1/5 F(n) - 3/5 F(n + 1) 10 4 3 2 4 3 - 13/2 F(n) F(n + 1) + 9/2 F(n) F(n + 1) + 115/3 F(n) F(n + 1) 6 41 979 5 2 + 59/3 F(n) F(n + 1) + -- F(n) + 4/15 F(n + 1) - --- F(n) F(n + 1) 30 30 7 11 7 23 3 - 25/6 F(n) + -- F(n + 1) - -- F(n + 1) 10 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, 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(3 n - j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 75 4 19 5 80 3 2 G(n) = -- F(n) - -- F(n) - 1/11 + 1/11 F(n + 1) - -- F(n) F(n + 1) 11 11 11 7 2941 2 2 87 2 3 + 325 F(n) F(n + 1) + ---- F(n) F(n + 1) + -- F(n) F(n + 1) 22 22 457 3 27 4 61 4 - --- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 11 22 11 4 4 7111 3 2 6 + 1375/2 F(n) F(n + 1) - ---- F(n) F(n + 1) - 1575/2 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, 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(3 n - j) F(2 n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 3 4 4 G(n) = 1/2 F(n) (229 F(n) F(n + 1) - 298 F(n) F(n + 1) 6 2 3 5 4 8 - 232 F(n) F(n + 1) + 159 F(n) F(n + 1) - 9 - 7 F(n) - 6 F(n + 1) 7 4 + 149 F(n) F(n + 1) + 15 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, 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 ----- \ 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) 5 4 2 3 3 G(n) = -20 F(n) F(n + 1) - 65 F(n) F(n + 1) + 30 F(n) F(n + 1) 3 4 6 - 30 F(n) F(n + 1) + 14 F(n) F(n + 1) + 41/2 F(n) F(n + 1) 2 6 2 5 + 11/2 F(n) F(n + 1) - 11/2 F(n) F(n + 1) + 29 F(n) F(n + 1) 3 2 2 7 + 1/2 F(n + 1) - 1/2 F(n + 1) - 7/2 F(n) + 1/2 F(n) 2 2 4 - 21/2 F(n) F(n + 1) + 35 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, 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(3 n - j) F(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 2 2 G(n) = 31/3 F(n) F(n + 1) + 5425/2 F(n) F(n + 1) + 1135/6 F(n) F(n + 1) 2 4 3 5 2 6 - 15 F(n) F(n + 1) - 2325 F(n) F(n + 1) - 75/2 F(n) F(n + 1) 2 5 2 8 - 5/3 F(n) - 75 F(n) F(n + 1) - 197/6 F(n + 1) - 2225/2 F(n + 1) 3 3 3 6 - 63 F(n) F(n + 1) + 80 F(n) F(n + 1) + 65/2 F(n + 1) 4 3 + 6637/6 F(n + 1) + 20/3 - 2849/6 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, 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(3 n - j) F(3 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 1157 2 6 G(n) = 120 F(n) F(n + 1) + ---- F(n + 1) - 105/2 F(n + 1) 22 3 3 2 7 10145 3 2 - 130 F(n) F(n + 1) - 625 F(n) F(n + 1) - ----- F(n) F(n + 1) 22 30981 5 9 32 2 13556 4 + ----- F(n + 1) - 5825/2 F(n + 1) + -- F(n) - ----- F(n) F(n + 1) 11 11 11 365 8 16073 2 3 - --- F(n) F(n + 1) + 14525/2 F(n) F(n + 1) + ----- F(n) F(n + 1) 22 22 2 4 3 6 120 + 45/2 F(n) F(n + 1) - 11325/2 F(n) F(n + 1) - --- F(n) 11 2111 + ---- 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, 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(3 n - j) F(4 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 7 3 4 G(n) = -60 F(n) + 1/2 + 6 F(n) - 1/2 F(n + 1) + 117/2 F(n) 2 2 2 3 4 - 25/2 F(n) F(n + 1) + 17/2 F(n) F(n + 1) + 7/2 F(n) F(n + 1) 5 2 7 5 3 + 143/2 F(n) F(n + 1) + F(n) F(n + 1) + 55 F(n) F(n + 1) 6 6 2 7 - 12 F(n) F(n + 1) + 77 F(n) F(n + 1) - 125 F(n) F(n + 1) 3 5 6 4 3 - 10 F(n) F(n + 1) + 1/2 F(n) F(n + 1) - 62 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, 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 ----- \ 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) 9 3 2 G(n) = -65/6 F(n) + 1/3 F(n + 1) - 1/3 F(n + 1) + 63 F(n) F(n + 1) 4 8 2 - 4555/2 F(n) F(n + 1) + 13721/6 F(n) F(n + 1) - 55/3 F(n) F(n + 1) 2 3 2 7 3 4 + 5705/6 F(n) F(n + 1) - 33227/6 F(n) F(n + 1) + 75/2 F(n) F(n + 1) 3 6 4 4 5 - 221/6 F(n) F(n + 1) + 217/2 F(n) F(n + 1) + 14626/3 F(n) F(n + 1) 6 2 5 3 2 - 125/2 F(n) F(n + 1) + 275/2 F(n) F(n + 1) - 356 F(n) F(n + 1) 4 3 3 - 325/2 F(n) F(n + 1) + 17/6 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, 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(3 n - j) F(5 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 9 3 G(n) = -1500 F(n + 1) + 3/2 F(n) + 2 F(n + 1) - 19/2 + 1263/2 F(n) F(n + 1) 4 3 5 2 6 - 1295/2 F(n) F(n + 1) + 3150 F(n) F(n + 1) + 25 F(n) F(n + 1) 7 8 2 2 - 3650 F(n) F(n + 1) + 3650 F(n) F(n + 1) - 499/2 F(n) F(n + 1) 2 3 2 7 3 2 + 243 F(n) F(n + 1) - 25 F(n) F(n + 1) - 127/2 F(n) F(n + 1) 3 6 8 4 - 3150 F(n) F(n + 1) + 1500 F(n + 1) - 1491 F(n + 1) 3 5 + 169/2 F(n) F(n + 1) + 2997/2 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, 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 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1125 4 1125 2 3 8242 3 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 2204 2204 551 3638 4 36225 3 5 6505 2 2 + ---- F(n) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 551 551 551 41875 7 26933 4 225 5 225 5 - ----- F(n) F(n + 1) - ----- F(n + 1) - ---- F(n) - ---- F(n + 1) 551 1102 1102 2204 7051 657 34425 8 775 2 6 - ---- - ---- F(n + 1) + ----- F(n + 1) + ---- F(n) F(n + 1) 1102 2204 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, 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(4 n - 4 j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 27 4 2 3 5 G(n) = -6/7 F(n) F(n + 1) + -- F(n) F(n + 1) - 3/7 F(n) F(n + 1) 28 4 4 5 6 2 + 16/7 F(n) F(n + 1) - 6/7 F(n) F(n + 1) + 4 F(n) F(n + 1) 5 2 4 + 3/14 F(n) F(n + 1) - 3/14 F(n + 1) + 3/14 F(n + 1) 7 27 2 4 3 3 15 8 - 3/7 F(n) F(n + 1) - -- F(n) F(n + 1) + 6/7 F(n) F(n + 1) + -- F(n) 28 28 17 2 6 5 3 15 6 + -- F(n) F(n + 1) - 6 F(n) F(n + 1) - -- F(n) 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, 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(4 n - 4 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 329 4 381 7 1039 8 2066 7 G(n) = --- F(n) + --- F(n) - ---- F(n) - ---- F(n) F(n + 1) 319 638 638 319 2286 2 5 1905 4 3 39 2 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - -- F(n) F(n + 1) 319 319 11 63 4 192 2 1426 2 2 - --- F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 319 319 11019 2 6 1905 6 7532 4 4 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 638 319 3689 5 3 2193 3 63 3 + ---- F(n) F(n + 1) + ---- 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, 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(4 n - 4 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 3 G(n) = -1/2 F(n + 1) (9 F(n) - 44 F(n) F(n + 1) - 300 F(n + 1) 2 5 7 2 + 5 F(n) F(n + 1) + 300 F(n + 1) + 130 F(n) F(n + 1) 6 4 3 - 730 F(n) F(n + 1) + 630 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, 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(4 n - 4 j) F(5 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 45 5 183 2 441 9911 7 2 G(n) = ---- F(n) F(n + 1) - ---- F(n) + ---- F(n + 1) + ---- F(n) F(n + 1) 551 2204 1102 5510 1541 8 6287 3 2 75 3 3 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 551 2755 1102 75 6 15 6 129 5 8739 5 4 - ---- F(n) - ---- F(n + 1) + ---- F(n) + ---- F(n) F(n + 1) 2204 2204 1102 1102 363 6 3 63 8 3067 2 3 - --- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 38 551 5510 497 2 7 867 2 - ---- F(n) F(n + 1) - ---- F(n + 1) 5510 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, 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 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(5 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3063 8 7431 2 7 3 6 G(n) = ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - 22/7 F(n) F(n + 1) 28 28 3303 4 5 4 39 2 + ---- F(n) F(n + 1) - 768/7 F(n) F(n + 1) - -- F(n) F(n + 1) 14 28 1259 2 3 9 3 + ---- F(n) F(n + 1) - 9/14 F(n) + 3/14 F(n + 1) + 9/14 F(n) 28 3 2 151 4 - 3/14 F(n + 1) + 9/28 F(n) F(n + 1) + --- F(n) F(n + 1) 28 229 3 2 - --- 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, 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(4 n - 4 j) F(5 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 51862 8 167 3 6 251213 2 7 G(n) = ------ F(n) F(n + 1) + --- F(n) F(n + 1) + ------ F(n) F(n + 1) 319 29 638 222863 4 5 3563 4 270 3 - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 638 638 319 384 4 63 9 219 3 104311 4 - --- F(n) - --- F(n + 1) - --- F(n) F(n + 1) + ------ F(n) F(n + 1) 319 638 319 638 15029 3 2 2 2 44017 2 3 + ----- F(n) F(n + 1) - 9/58 F(n) F(n + 1) - ----- F(n) F(n + 1) 638 638 63 705 5 + --- + --- 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, 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(4 n - 4 j) F(5 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 6 5 G(n) = 1/10 F(n) F(n + 1) (-29 F(n + 1) + 70 F(n) - 247 F(n + 1) F(n) 3 3 2 4 - 240 F(n + 1) F(n) + 450 F(n + 1) F(n) - 23 F(n + 1) F(n) 6 + 19 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(4 n - 4 j) F(6 n - 6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 378 10 33138 9 101751 5 G(n) = ----- F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 3781 94525 94525 6804 4 76402 10 67102 2 372 - ----- F(n) F(n + 1) - ----- F(n) + ----- F(n) + ---- F(n) 18905 94525 94525 3781 378 9 21019 9 1512 7 2 + ---- F(n + 1) - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 3781 94525 3781 378 8 1142594 7 3 144671 8 2 + ---- F(n) F(n + 1) + ------- F(n) F(n + 1) - ------ F(n) F(n + 1) 3781 94525 189050 7938 6 3 756 8 1134 2 7 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 18905 18905 3781 789696 6 4 525937 2 4 4158 2 3 - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 94525 189050 18905 131464 - ------ F(n) F(n + 1) 94525 Proof: Both 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(4 n - 4 j) F(6 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 909 2 5 9696 5 11882 6 4 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 1102 551 551 4545 3 4 4545 5 2 6056 4 6 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 2204 2204 551 6943 9 48399 8 2 1050 9 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 551 1102 551 4083 2 8 618 2 4 15604 7 3 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) + ----- F(n) F(n + 1) 1102 551 551 909 7 48 6 441 10 1101 3 + ---- F(n) + --- F(n) + ---- F(n + 1) - ---- F(n) 2204 551 1102 2204 195 2 441 3 + ---- F(n) 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, 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(4 n - 4 j) F(6 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 13375 6 33 2 3 7 G(n) = ----- F(n + 1) + -- F(n) - 13750/7 F(n) F(n + 1) 14 28 3 5375 5 + 33/7 F(n) F(n + 1) - 64/7 F(n) F(n + 1) - ---- F(n) F(n + 1) 14 5625 2 8 3 3 9 - ---- F(n) F(n + 1) - 1375/7 F(n) F(n + 1) + 2500 F(n) F(n + 1) 28 2 2 7125 2 4 27 3 - 15/4 F(n) F(n + 1) + ---- F(n) F(n + 1) - -- F(n) F(n + 1) 28 14 28125 10 27 4 1381 2 33 - ----- F(n + 1) + -- F(n + 1) + ---- F(n + 1) - -- 28 28 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, 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(4 n - 4 j) F(6 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 543 6 3107 549 63 10 G(n) = ---- F(n) + ---- F(n) F(n + 1) - ---- F(n) - --- F(n + 1) 319 1276 5104 638 9237 5 63 9 15435 4 6 513 8 + ---- F(n) + --- F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 5104 638 638 5104 2115 9 15425 4 2 10233 5 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 1276 319 1276 7995 5 4 5109 4 5 267 8 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 1276 1276 88 175315 7 3 34385 6 4 10965 6 3 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 1276 638 2552 20813 5 5 1817 5 3057 4 - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 2552 28497 7 2 - ----- F(n) F(n + 1) 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, 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(4 n - 4 j) F(6 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 6 3 4 G(n) = -1/8 F(n + 1) (-6 F(n + 1) - 60 F(n + 1) F(n) + 205 F(n + 1) F(n) 5 4 5 4 5 - 228 F(n + 1) F(n) - 302 F(n) + 1036 F(n + 1) F(n) 3 6 9 9 - 1045 F(n + 1) F(n) + 394 F(n) + 3 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, 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 ----- \ 2 G(n) = ) F(j) F(4 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = 1/6 - 1/6 F(n + 1) + 1/6 F(n) F(n + 1) - 7/12 F(n) 2 2 3 4 + 5/3 F(n) F(n + 1) - 5/3 F(n) F(n + 1) + 5/12 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, 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 ----- \ 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) 63 63 3 345 2 287 3 G(n) = --- - --- F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 638 638 638 319 453 2 21 2 2 105 3 259 3 - --- F(n) F(n + 1) - -- F(n) F(n + 1) + --- F(n) + --- F(n) F(n + 1) 638 58 319 319 273 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, 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(4 n - 4 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 441 441 183 21 3 G(n) = - ---- + ---- F(n + 1) + ---- F(n) - --- F(n) F(n + 1) 1102 1102 1102 551 4347 2 2 3423 2 3 2163 3 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 1102 1102 551 837 3 2 1071 4 78 4 1329 5 - ---- F(n) F(n + 1) - ---- F(n) - -- F(n) F(n + 1) + ---- F(n) 1102 1102 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, 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(4 n - 4 j) F(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1215 2 15 2 1425 2 4 225 3 G(n) = ---- F(n + 1) + -- F(n) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 77 22 154 77 705 75 3 375 2 2 15 - --- F(n) F(n + 1) - -- F(n) F(n + 1) - --- F(n) F(n + 1) - -- 154 77 154 22 6225 3 3 5475 5 1200 6 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n + 1) 154 154 77 75 4 + --- F(n + 1) 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, 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 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 28275 2 378 3 4200 3 378 G(n) = ------ F(n) F(n + 1) - ---- F(n + 1) - ---- F(n) + ---- 7562 3781 3781 3781 3836 3 13017 2 9534 2 2 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 3781 3781 3781 11452 3 59265 3 4 3822 4 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) 3781 3781 3781 39465 6 124785 2 5 18705 4 3 + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 7562 7562 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, 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(4 n - 4 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 2 G(n) = -1/12 (-2 F(n + 1) + F(n)) (7 F(n) - 13 F(n + 1) F(n) - 8 F(n) 2 2 2 + 13 F(n + 1) F(n) + F(n) F(n + 1) + 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, 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(4 n - 4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 63 63 359 615 3 G(n) = - --- + --- F(n + 1) + --- F(n) - --- F(n) F(n + 1) 638 638 319 638 621 2 2 953 2 3 531 3 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 638 638 638 1239 3 2 207 4 2789 4 241 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, 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(4 n - 4 j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7137 2 3 3495 2 4 36469 3 2 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 551 1102 1102 132 5 33 13301 4 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - ----- F(n) F(n + 1) 29 551 551 2085 5 2445 6 2445 6307 - ---- F(n) F(n + 1) - ---- F(n) + ---- F(n) - ---- F(n + 1) 551 1102 1102 551 1587 2 12173 5 1014 6 - ---- F(n + 1) + ----- F(n + 1) + ---- F(n + 1) 1102 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, 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(4 n - 4 j) F(n + 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 171 5 687 6 111 2 15 3 G(n) = --- F(n) + --- F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n + 1) 154 308 44 77 171 7 15 4 244 2 3 - --- F(n) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 154 154 77 4107 3 4 3183 5 2 48 3 2 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + -- F(n) F(n + 1) 308 308 11 73 4 4 3 450 6 - -- F(n) F(n + 1) + 141/7 F(n) F(n + 1) + --- F(n) F(n + 1) 22 77 15 - -- 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, 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(4 n - 4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 3 3 G(n) = 131/6 F(n + 1) - 5/4 + 5/4 F(n) - 160/3 F(n) F(n + 1) 5 13 4 3 + 50 F(n) F(n + 1) + -- F(n + 1) + 29/6 F(n) F(n + 1) 12 3 2 2 - 41/6 F(n) F(n + 1) - 13/6 F(n) F(n + 1) - 15/4 F(n) F(n + 1) 2 4 6 + 10 F(n) F(n + 1) - 65/3 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, 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(4 n - 4 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3969 2 687 5 2328 6 63 63 2 G(n) = ---- F(n) + --- F(n) - ---- F(n) - --- F(n + 1) + --- F(n + 1) 638 638 319 638 638 645 2 3 2 4 243 3 2 - --- F(n) F(n + 1) - 15/2 F(n) F(n + 1) + --- F(n) F(n + 1) 638 319 5820 3 3 1539 4 2630 5 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 319 69 4 431 5 - -- F(n) 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, 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(4 n - 4 j) F(2 n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 639 2 128310 6 2 4 G(n) = ---- F(n) F(n + 1) - ------ F(n) F(n + 1) - 30 F(n) F(n + 1) 38 551 20835 2 5 32010 3 3 205065 3 4 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 1102 551 1102 11230 5 13355 6 103755 7 + ----- F(n) F(n + 1) + ----- F(n + 1) + ------ F(n + 1) 551 551 1102 26500 5 48831 2 3957 2 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) 551 1102 1102 1429 2 3957 3 51657 3 - ---- F(n + 1) + ---- F(n) - ----- F(n + 1) 58 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, 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(4 n - 4 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 23 3 2 113 2 3 G(n) = 217/2 F(n + 1) - -- F(n) + 49/4 F(n) F(n + 1) - --- F(n) F(n + 1) 12 12 2 5 313 3 2 7 + 50/3 F(n) F(n + 1) + --- F(n) F(n + 1) - 325/3 F(n + 1) 12 4 503 2 6 - 61/3 F(n) F(n + 1) - --- F(n) F(n + 1) + 775/3 F(n) F(n + 1) 12 3 4 113 23 5 - 725/3 F(n) F(n + 1) - --- F(n + 1) + -- F(n) + 37/4 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, 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 ----- \ G(n) = ) F(j) F(4 n - 4 j) F(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 288 2 4 1176 3 3 4341 4 2 G(n) = --- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 638 63 3 2616 5 63 2 582 2 + --- F(n + 1) + ---- F(n) F(n + 1) - --- F(n + 1) - --- F(n) 638 319 638 319 582 3 903 5 6 + --- F(n) - --- F(n) F(n + 1) - 75/2 F(n) F(n + 1) 319 638 7193 2 2 5 3 4 - ---- F(n) F(n + 1) + 175/2 F(n) F(n + 1) + 25 F(n) F(n + 1) 638 4 3 24931 2 - 100 F(n) F(n + 1) + ----- 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, 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 ----- \ 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) 2 4 2 6 3 G(n) = -19/6 F(n) + 3 F(n) + 1/6 - 1525/3 F(n) F(n + 1) - 22 F(n) F(n + 1) 3 3 4 4 + 125/6 F(n) F(n + 1) + 500 F(n) F(n + 1) + 101/6 F(n) F(n + 1) 7 2 2 2 + 650/3 F(n) F(n + 1) + 241/3 F(n) F(n + 1) - 1/6 F(n + 1) 625 4 2 2 4 5 - --- F(n) F(n + 1) + 125/4 F(n) F(n + 1) - 50/3 F(n) F(n + 1) 12 3 5 3 - 50 F(n) F(n + 1) - 650/3 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, 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(4 n - 4 j) F(4 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 14334 2 29715 6 66735 2 5 G(n) = ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 638 638 34185 4 3 63 3 3 - ----- F(n) F(n + 1) - --- F(n + 1) + 202 F(n) F(n + 1) 319 638 7 9849 2 2 2 - 200 F(n) F(n + 1) - ---- F(n) F(n + 1) - 74 F(n) F(n + 1) 638 2 6 3 5895 3 4 + 475 F(n) F(n + 1) + 20 F(n) F(n + 1) + ---- F(n) F(n + 1) 638 4 4 3 5 63 4 1851 3 - 925/2 F(n) F(n + 1) + 50 F(n) F(n + 1) + --- - 3 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, 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(4 n - 4 j) F(5 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 497 2 85 8 41 6 G(n) = ---- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 60 24 60 4 3 5 2 4 5 - 41 F(n) F(n + 1) + 205/4 F(n) F(n + 1) + 275/3 F(n) F(n + 1) 733 6 4577 7 2 2627 6 3 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 60 24 12 4 8 155 + 37/4 F(n) F(n + 1) + 1289/4 F(n) F(n + 1) - --- F(n) 24 13 2 3 9 9 - -- F(n) F(n + 1) - 1/6 F(n + 1) + 1/6 F(n + 1) + 271/2 F(n) 30 61 7 5 + -- F(n) - 1073/8 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, 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 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 128 8 288 3 4832 2 6 2906 3 5 G(n) = --- F(n + 1) - --- F(n) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 319 319 387 4 4 2392 3 189 2 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 29 319 319 300 2 31 4 4183 6 2 + --- F(n) F(n + 1) - --- F(n) - ---- F(n) F(n + 1) 319 319 319 2840 7 1358 3 128 3 - ---- 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, 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(4 n - 3 j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 6 3 5 G(n) = 65/2 - 63/2 F(n) - 185 F(n) F(n + 1) + 50 F(n) F(n + 1) 4 4 2 2 3 + 365/2 F(n) F(n + 1) + 111/2 F(n) F(n + 1) - 143/2 F(n) F(n + 1) 8 4 + 65/2 F(n + 1) - 65 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, 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(4 n - 3 j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 931 3 925 7 4375 2 6 G(n) = --- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 11 22 4275 4 4 15 27 4 - ---- F(n) F(n + 1) - 1/11 - -- F(n) + 1/11 F(n + 1) - -- F(n) 22 22 22 225 3 5 15 4 685 2 2 + --- F(n) F(n + 1) + -- F(n) F(n + 1) - --- F(n) F(n + 1) 11 22 22 32 2 3 93 3 21 3 2 - -- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 11 11 22 49 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, 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(4 n - 3 j) F(5 n - 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 865 55 2 35215 5 4 1910 4 G(n) = ---- F(n) + --- F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 1102 551 551 551 10755 3 2 680 3 3 14275 2 3 + ----- F(n) F(n + 1) + --- F(n) F(n + 1) - ----- F(n) F(n + 1) 1102 551 551 15 10825 4 5 68075 3 6 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 1102 1102 1102 15050 2 7 185 5 340 6 340 2 4 + ----- F(n) F(n + 1) - ---- F(n) - --- F(n) - --- F(n) F(n + 1) 551 1102 551 551 340 4 2 680 5 55 9 + --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n + 1) 551 551 551 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, 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(4 n - 3 j) F(5 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 3 5 27 5 G(n) = -3/14 F(n + 1) - 5/14 F(n) - 9/14 F(n) + -- F(n + 1) 56 825 4 5 1545 5 4 2 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) + 3/4 F(n) F(n + 1) 56 14 11 2 2 3 2475 2 7 - -- F(n) F(n + 1) + 185/4 F(n) F(n + 1) - ---- F(n) F(n + 1) 28 56 15 9 15 4 3 15 2 5 - -- F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 56 14 14 3 2 115 4 3 6 - 110/7 F(n) F(n + 1) + --- F(n) F(n + 1) + 425/4 F(n) F(n + 1) 28 5 2 - 3/7 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, 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 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(5 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7936 4 5 2308 3 6 289 8 193 9 G(n) = ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n + 1) + --- F(n) 319 319 638 319 897 4 339 4 124 5 645 + --- F(n) + --- F(n + 1) + --- F(n + 1) + 4/319 F(n + 1) - --- 638 319 319 638 1445 3 5 2280 4 1734 5 3 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 8475 5 4 1445 6 2 3959 8 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 319 340 3 1138 3 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, 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 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(5 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 39 5 63 3 3 337 4 49 5 G(n) = -- F(n) + -- F(n) F(n + 1) - --- F(n) F(n + 1) - -- F(n) F(n + 1) 22 22 22 11 3683 2 3 3657 2 7 1301 3 2 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 22 22 22 3 6 1257 4 5 4392 5 4 - 769/2 F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 22 11 8 159 2 4 + 9/22 F(n) F(n + 1) - --- F(n) F(n + 1) - 1/11 F(n + 1) 44 67 4 2 6 27 2 2 + -- F(n) F(n + 1) + 3/44 F(n + 1) + -- F(n) + 1/44 F(n + 1) 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, 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(4 n - 3 j) F(6 n - 6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 5 G(n) = -5/3 F(n) F(n + 1) (-2 F(n + 1) F(n) - 9 F(n + 1) F(n) 4 2 6 6 2 + 10 F(n) F(n + 1) + 2 F(n) - F(n + 1) F(n) + 4 F(n + 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, 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(4 n - 3 j) F(6 n - 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 48 3 65970 2 4 775985 2 8 G(n) = ---- F(n) - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 551 551 1102 31490 3 3 15855 4 2 646235 4 6 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 551 551 1102 3208 155285 5 8310 9 + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 551 551 29 179 2 55 10 14235 3 7 503 2 - --- F(n) F(n + 1) - --- F(n + 1) - ----- F(n) F(n + 1) - --- F(n) 551 551 551 551 55 3 178 2 + --- F(n + 1) - --- F(n) F(n + 1) 551 551 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, 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(4 n - 3 j) F(6 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 5 33 2 1369 3 3 G(n) = 1/28 F(n) - 3378/7 F(n) F(n + 1) + -- F(n) - ---- F(n) F(n + 1) 28 14 631 3 7 1381 4 2 28131 4 6 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 14 28 28 6869 9 11 2 2 - 64/7 F(n) F(n + 1) + ---- F(n) F(n + 1) - -- F(n) F(n + 1) 14 14 2 8 3 10 - 8439/7 F(n) F(n + 1) - 3/14 F(n) F(n + 1) + 3/14 F(n + 1) - 3/14 2 4 + 1436/7 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, 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(4 n - 3 j) F(6 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20750 3 7 224 2 116101 6 10543 2 G(n) = ------ F(n) F(n + 1) - --- F(n) + ------ F(n + 1) - ----- F(n + 1) 319 319 319 29 78 5 206 414 5 16375 4 6 + --- F(n + 1) - --- F(n + 1) - --- F(n) - ----- F(n) F(n + 1) 319 319 319 11 4102 5 91 2 3 221355 2 4 + ---- F(n) F(n + 1) - -- F(n) F(n + 1) - ------ F(n) F(n + 1) 319 29 319 570250 2 8 601 3 2 274550 3 3 + ------ F(n) F(n + 1) + --- F(n) F(n + 1) + ------ F(n) F(n + 1) 319 319 319 26 4 101080 4 2 232250 9 + -- F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 29 319 319 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, 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(4 n - 3 j) F(6 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 65 2 5 2515 2 16125 2 8 G(n) = --- F(n) F(n + 1) - ---- F(n + 1) + ----- F(n) F(n + 1) 44 22 22 78500 3 7 353 5 99875 9 + ----- F(n) F(n + 1) + --- F(n) F(n + 1) - ----- F(n) F(n + 1) 11 11 11 94 2 20325 2 4 479 5 2 + -- F(n) F(n + 1) - ----- F(n) F(n + 1) - --- F(n) F(n + 1) 11 22 22 34 2 6 21 3 453 3 - -- F(n) - 7067/2 F(n + 1) - -- F(n) + --- F(n + 1) 11 11 44 17115 5 40125 10 449 7 + ----- F(n) F(n + 1) + ----- F(n + 1) - --- F(n + 1) 11 11 44 455 6 6085 3 3 355 4 3 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 22 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, 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 ----- \ 2 G(n) = ) F(j) F(4 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 14 3 G(n) = 1/11 - 1/11 F(n + 1) - 6/11 F(n) F(n + 1) + -- F(n) F(n + 1) 11 2 2 2 3 18 3 - 3/11 F(n) F(n + 1) - F(n) F(n + 1) - 4/11 F(n) + -- F(n) F(n + 1) 11 4 - 8/11 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, 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 ----- \ 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) 128 128 76 32 3 G(n) = - --- + --- F(n + 1) + --- F(n) + --- F(n) F(n + 1) 319 319 319 319 1328 2 2 103 2 3 1344 3 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 29 319 81 3 2 344 4 20 4 22 5 - -- F(n) F(n + 1) - --- F(n) + -- F(n) F(n + 1) - -- F(n) 29 319 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, 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 ----- \ 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) 747 2 2 45 3 G(n) = --- F(n + 1) + 15/4 F(n) + -- F(n) F(n + 1) - 128/7 F(n) F(n + 1) 14 14 15 3 75 2 2 237 2 4 - -- F(n) F(n + 1) - -- F(n) F(n + 1) + --- F(n) F(n + 1) 14 28 14 1741 5 1767 3 3 15 4 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + -- F(n + 1) - 3/4 14 14 28 6 - 372/7 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, 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 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 86035 2 590 190565 3 16175 6 G(n) = ------ F(n) F(n + 1) + --- + ------ F(n + 1) + ----- F(n) F(n + 1) 1102 551 1102 38 6700 3 4 6575 7 540 3 - ---- F(n) F(n + 1) - ---- F(n + 1) + --- F(n) F(n + 1) 19 38 551 15810 2 1880 2 2 325 2 5 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 551 551 19 2680 3 3345 3 535 4 - ---- F(n) F(n + 1) - ---- F(n) - --- F(n + 1) 551 551 551 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, 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 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 6 4 8 G(n) = -74 F(n) F(n + 1) + 880 - 872 F(n) - 880 F(n + 1) 3 2 2 3 + 1361 F(n) F(n + 1) + 1038 F(n) F(n + 1) - 1808 F(n) F(n + 1) 3 5 7 - 1804 F(n) F(n + 1) + 2159 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, 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 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 31 19 3 G(n) = - 1/11 + 1/11 F(n + 1) + -- F(n) - -- F(n) F(n + 1) 22 22 15 2 2 16 2 3 13 3 - -- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 22 97 3 2 4 53 4 19 5 + -- F(n) F(n + 1) + 6/11 F(n) - -- F(n) F(n + 1) - -- F(n) 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, 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 ----- \ 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) 1611 5 668 4188 2 3 G(n) = ----- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 298 2 4 2914 3 2 2018 3 3 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 29 319 319 2912 452 2 2272 5 324 6 - ---- F(n + 1) + --- F(n + 1) + ---- F(n + 1) - --- F(n + 1) 1595 319 1595 319 4104 7529 5 272 6 - ---- F(n) + ---- F(n) + --- 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, 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 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 829 4 3 1063 5 2 331 2 5 G(n) = --- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 56 28 56 3 4 27 5 933 6 - 64/7 F(n) F(n + 1) - -- F(n + 1) + --- F(n) F(n + 1) 56 28 7 37 2 3 129 3 2 - 9/56 F(n + 1) - -- F(n) F(n + 1) + --- F(n) F(n + 1) 56 28 4 15 15 85 3 3 - 47/8 F(n) F(n + 1) + -- F(n) + -- F(n + 1) - -- F(n) + 3/8 F(n + 1) 14 56 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, 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 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(n + 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 815 3 5 10931 4 2005 55 G(n) = --- F(n) F(n + 1) + ----- F(n) - ---- F(n) + --- F(n + 1) 38 1102 1102 551 302 4 85331 2 2 222 2 3 + --- F(n) F(n + 1) + ----- F(n) F(n + 1) + --- F(n) F(n + 1) 551 551 551 57657 3 9917 3 2 4986 4 55 - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- 1102 1102 551 551 394619 3 6835 7 16690 2 6 - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 1102 19 19 14130 4 4 + ----- F(n) F(n + 1) 19 Proof: Both 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 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 13 2 5 G(n) = -F(n) F(n + 1) - 1/11 F(n + 1) + 1/11 F(n + 1) - -- F(n) - 9/11 F(n) 11 17 5 21 2 3 13 2 4 + -- F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 11 22 22 27 3 2 38 3 3 25 4 - -- F(n) F(n + 1) + -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 22 76 4 2 72 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, 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 ----- \ G(n) = ) F(j) F(4 n - 3 j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6524 865 6 93 2 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) + -- F(n) F(n + 1) 319 29 11 1760 2 5 3 3 695 3 4 - ---- F(n) F(n + 1) + 30 F(n) F(n + 1) - --- F(n) F(n + 1) 29 29 2345 4 3 128 2 1157 2 438 3 + ---- F(n) F(n + 1) - --- F(n + 1) - ---- F(n) - --- F(n) 29 319 319 319 128 3 9551 2 5 + --- F(n + 1) - ---- F(n) F(n + 1) - 20 F(n) F(n + 1) 319 319 2 4 4 2 + 35 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, 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(4 n - 3 j) F(2 n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 51 2 2 6 32401 4 G(n) = --- F(n) - 229/7 F(n + 1) + 65/2 F(n + 1) + ----- F(n + 1) 28 28 3 5 995 2 6 - 3513/7 F(n) F(n + 1) - 75 F(n) F(n + 1) - --- F(n) F(n + 1) 14 143 39965 7 5947 2 2 + --- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 14 14 28 2 4 3 5 16335 8 275 - 15 F(n) F(n + 1) - 16890/7 F(n) F(n + 1) - ----- F(n + 1) + --- 14 28 1115 3 3 3 - ---- F(n) F(n + 1) + 80 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, 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(4 n - 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 97 5 149 2 4 G(n) = 1/11 F(n + 1) + -- F(n) F(n + 1) - --- F(n) F(n + 1) 22 11 431 2 311 4 2 14 2 2 + --- F(n) F(n + 1) + --- F(n) F(n + 1) + -- F(n) - 1/11 F(n + 1) 11 22 11 6 125 2 2 5 - 75/2 F(n) F(n + 1) - --- F(n) F(n + 1) + 175/2 F(n) F(n + 1) 11 58 3 3 3 4 4 3 - -- F(n) F(n + 1) + 25 F(n) F(n + 1) - 100 F(n) F(n + 1) 11 61 19 3 - -- 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, 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(4 n - 3 j) F(3 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 16580 5 3 14739 3 582 8 16581 7 G(n) = ------ F(n) F(n + 1) - ----- F(n) + --- F(n) + ----- F(n) 319 319 319 319 164 7 4172 2 5 21415 6 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 319 3060 2 6 1524 3 5 3 4 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + 75 F(n) F(n + 1) 319 319 6115 3 2235 7 19908 6 2 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 319 200 6 44940 4 3 128 128 3 + --- F(n) F(n + 1) - ----- F(n) F(n + 1) + --- - --- F(n + 1) 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, 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(4 n - 3 j) F(4 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 57 3 4 8 23 7 159 5 2 G(n) = --- F(n) F(n + 1) - 2 F(n) - -- F(n) - --- F(n) F(n + 1) 11 11 22 2 5 10 4 7 3 - 8/11 F(n) F(n + 1) - -- + F(n + 1) - 7/11 F(n + 1) + 6/11 F(n + 1) 11 3 5 4 4 43 4 3 + 22 F(n) F(n + 1) - 67/2 F(n) F(n + 1) + -- F(n) F(n + 1) 22 7 6 2 127 6 + 18 F(n) F(n + 1) - 29 F(n) F(n + 1) + --- F(n) F(n + 1) 22 5 3 + 32 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(4 n - 3 j) F(5 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 415 2 6 9 51 494 37 G(n) = ---- F(n) F(n + 1) + 4075/2 F(n + 1) + -- F(n) - --- F(n + 1) + -- 11 11 11 11 2 3 2913 3 8080 7 - 435 F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 22 11 553 3 657 2 2 3 6 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + 8225/2 F(n) F(n + 1) 22 11 13445 3 5 2 7 3 2 - ----- F(n) F(n + 1) + 525/2 F(n) F(n + 1) + 235 F(n) F(n + 1) 22 4 8 5 + 1715/2 F(n) F(n + 1) - 5025 F(n) F(n + 1) - 3985/2 F(n + 1) 6489 4 6565 8 + ---- F(n + 1) - ---- 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, 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 ----- \ 2 G(n) = ) F(j) F(4 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 72 3 4 G(n) = -1/22 F(n) F(n + 1) + -- F(n) F(n + 1) + 4/11 F(n) F(n + 1) 11 14 3 2 4 7 - -- F(n) F(n + 1) - 1/2 F(n) F(n + 1) - 6/11 F(n) F(n + 1) 11 287 6 2 23 4 4 17 2 6 - --- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 11 22 11 5 27 8 7 252 5 3 - 2/11 F(n) - -- F(n) + 6/11 F(n) F(n + 1) + --- F(n) F(n + 1) 22 11 - 9/22 F(n + 1) + 9/22 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, 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 ----- \ 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) 2 4 8 8 6 G(n) = -13/3 F(n) F(n + 1) + 1/3 F(n + 1) + 10/3 F(n) - 4/3 F(n) 6 5 5 6 2 - 1/3 F(n + 1) + F(n) F(n + 1) + 3 F(n) F(n + 1) + 34 F(n) F(n + 1) 5 3 2 6 4 4 - 128/3 F(n) F(n + 1) + 6 F(n) F(n + 1) + 95/3 F(n) F(n + 1) 3 5 7 3 3 - 40/3 F(n) F(n + 1) - 4/3 F(n) F(n + 1) + 4 F(n) F(n + 1) 7 4 2 - 44/3 F(n) F(n + 1) - 16/3 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, 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(4 n - 2 j) F(5 n - 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 9 65 9 6 41 6 G(n) = -5/42 F(n + 1) - -- F(n) + 4/21 F(n) F(n + 1) + -- F(n) F(n + 1) 14 42 340 7 2 20 2 5 6 3 - --- F(n) F(n + 1) - -- F(n) F(n + 1) + 67/2 F(n) F(n + 1) 21 21 10 7 3 4 11 2 + 25/6 F(n) + -- F(n) + 5/21 F(n) F(n + 1) - -- F(n) F(n + 1) 21 12 2 3 17 2 4 + 59/6 F(n) F(n + 1) - -- F(n) F(n + 1) - 8 F(n) F(n + 1) 28 1135 8 125 2 7 367 8 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 84 14 84 7 + 5/42 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, 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(4 n - 2 j) F(5 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 193 5 126 4 63 4 14879 4 5 G(n) = --- F(n) + --- F(n) + --- F(n + 1) - ----- F(n) F(n + 1) 319 319 638 638 51996 5 4 134 8 481 2 2 + ----- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 638 43379 2 3 7674 3 2 4201 4 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 319 638 14 3 43229 2 7 50159 3 6 + --- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 638 319 63 9 245 3 - --- 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, 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(4 n - 2 j) F(5 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 8727 4 343 3 6 107 4 2 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 22 22 22 38 2 4 12 5 2 2 - -- F(n) F(n + 1) + -- F(n) F(n + 1) - 9/22 F(n + 1) + 5/22 F(n) 11 11 16 8793 8 1807 2 3 - -- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 22 11 10608 2 7 679 3 2 41 3 3 + ----- F(n) F(n + 1) + --- F(n) F(n + 1) - -- F(n) F(n + 1) 11 11 11 203 4 9408 4 5 39 9 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) + -- F(n) + 9/22 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, 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(4 n - 2 j) F(5 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 85 8 3 6 1283 4 G(n) = -- F(n) F(n + 1) - 115/3 F(n) F(n + 1) + ---- F(n) F(n + 1) 18 36 1357 4 5 6 1585 6 3 + ---- F(n) F(n + 1) - 13/2 F(n) F(n + 1) + ---- F(n) F(n + 1) 36 12 2725 7 2 125 8 6 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) + 21/2 F(n) F(n + 1) 18 12 5 2 3 + 10 F(n) F(n + 1) + 25/9 F(n + 1) + 14/3 F(n + 1) 2 5 5 4 3 - 11/6 F(n) F(n + 1) - 22/9 F(n + 1) - 115/6 F(n) F(n + 1) 7 7 9 + 13/6 F(n) - 5 F(n + 1) - 31/6 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, 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(4 n - 2 j) F(6 n - 6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 646125 10 54 3 173854 2 2778 3 G(n) = ------ F(n + 1) + ---- F(n + 1) - ------ F(n + 1) + ---- F(n) 1102 6061 6061 6061 2778 2 64875 2 8 63200 3 3 - ---- F(n) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 6061 551 551 856 2 614525 6 81825 2 4 - ---- F(n) F(n + 1) - ------ F(n + 1) - ----- F(n) F(n + 1) 6061 1102 551 632000 3 7 123575 5 804125 9 + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 551 551 551 4724 2 30880 - ---- 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, 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 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(6 n - 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 17 4 20 2 2 4315 2 4 G(n) = 5/42 + -- F(n) - -- F(n) F(n + 1) + ---- F(n) F(n + 1) 84 21 21 25310 2 8 3 4135 3 3 - ----- F(n) F(n + 1) - 1/21 F(n) F(n + 1) - ---- F(n) F(n + 1) 21 42 4115 4 2 84365 4 6 185 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - --- F(n) F(n + 1) 84 84 21 20635 9 10 1865 3 7 19 2 + ----- F(n) F(n + 1) - 5/42 F(n + 1) + ---- F(n) F(n + 1) + -- F(n) 42 42 28 10120 5 - ----- F(n) 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, 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(4 n - 2 j) F(6 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1389 5 40243 2 3 3568 9 139306 10 G(n) = ---- F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) - ------ F(n + 1) 1595 14355 4785 13079 16948 9 767 10 9071 11729 - ----- F(n + 1) - --- F(n) + ---- F(n) + ----- F(n + 1) 14355 638 9570 28710 296849 6 10410 2 98555 8 2 + ------ F(n + 1) - ----- F(n + 1) - ----- F(n) F(n + 1) 26158 13079 26158 43708 8 624900 7 3 45046 7 2 - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 14355 13079 14355 168425 9 1943245 6 4 282875 9 + ------ F(n) F(n + 1) - ------- F(n) F(n + 1) + ------ F(n) F(n + 1) 13079 26158 13079 38669 5 31666 8 16 4 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - -- F(n) F(n + 1) 13079 14355 87 63137 - ----- F(n) F(n + 1) 13079 Proof: Both 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(4 n - 2 j) F(6 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 40125 10 278 3 76325 6 215 2 G(n) = ----- F(n + 1) + --- F(n + 1) - ----- F(n + 1) - --- F(n) F(n + 1) 11 11 22 22 16125 2 8 7850 3 3 3 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - 9/22 F(n) 22 11 1325 3 4 15350 5 670 6 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) + --- F(n) F(n + 1) 22 11 11 99875 9 353 65 2 5 - ----- F(n) F(n + 1) + --- F(n) F(n + 1) + -- F(n) F(n + 1) 11 11 22 20325 2 4 45 2 78500 3 7 - ----- F(n) F(n + 1) + -- F(n) F(n + 1) + ----- F(n) F(n + 1) 22 22 11 565 7 57 2 2 - --- F(n + 1) - -- F(n) - 178 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, 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(4 n - 2 j) F(6 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 8 10 G(n) = 506/3 F(n) F(n + 1) - 1875 F(n) F(n + 1) - 9375 F(n + 1) 4 2 3 6 - 2431/6 F(n + 1) + 17/2 F(n) + 53/2 F(n) F(n + 1) + 26750/3 F(n + 1) 8 5 7 + 1225/3 F(n + 1) - 10750/3 F(n) F(n + 1) - 2975/3 F(n) F(n + 1) 9 2 2 2 4 + 70000/3 F(n) F(n + 1) - 70 F(n) F(n + 1) + 2375 F(n) F(n + 1) 3 3 3 5 - 5500/3 F(n) F(n + 1) + 2575/3 F(n) F(n + 1) 3 7 2 - 55000/3 F(n) F(n + 1) + 1376/3 F(n + 1) - 251/3 F(n) F(n + 1) - 7/2 Proof: Both 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 ----- \ 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) 3 G(n) = - 9/22 + 9/22 F(n + 1) + 5/22 F(n) - 3/11 F(n) F(n + 1) 81 2 2 2 3 39 3 - -- F(n) F(n + 1) + 4 F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 3 2 4 4 - 4 F(n) F(n + 1) - 9/11 F(n) + 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, 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(4 n - 2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 3 5 G(n) = -5/3 F(n) F(n + 1) + 5/3 F(n) F(n + 1) - 47/3 F(n) F(n + 1) - 1/6 2 31 2 4 2 4 + 1/6 F(n + 1) + -- F(n) - 5/12 F(n) + 11/6 F(n) F(n + 1) 12 3 3 4 2 5 - 35/6 F(n) F(n + 1) + 53/3 F(n) F(n + 1) - 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, 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(4 n - 2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 63 3 342 4 1062 6 63 2661 7 G(n) = ---- F(n + 1) + --- F(n) + ---- F(n) F(n + 1) + --- - ---- F(n) 638 319 1595 638 638 2531 2 351 3 9703 2 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 3190 319 3190 153 2 2 1017 3 537 4 3 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 58 319 638 5 2 28694 6 - 25/2 F(n) F(n + 1) + ----- 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, 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(4 n - 2 j) F(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 2 7 G(n) = 5/2 F(n) (-346 F(n) F(n + 1) + 55 F(n) F(n + 1) + 146 F(n + 1) 3 3 4 2 5 - 146 F(n + 1) + 330 F(n) F(n + 1) - 26 F(n) F(n + 1) 2 3 - 15 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, 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(4 n - 2 j) F(6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4200 3 1456450 3 2 108 3 5 G(n) = ----- F(n) F(n + 1) - ------- F(n) F(n + 1) + ---- F(n) F(n + 1) 6061 6061 6061 31297 3 6 475394 4 35993967 4 5 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + -------- F(n) F(n + 1) 551 6061 12122 17153809 4 17260291 8 3312 2 2 - -------- F(n) F(n + 1) + -------- F(n) F(n + 1) + ---- F(n) F(n + 1) 12122 12122 6061 7372157 2 3 21048146 2 7 54 9 + ------- F(n) F(n + 1) - -------- F(n) F(n + 1) - ---- F(n + 1) 12122 6061 6061 1602 4 108 7 54 2 6 + ---- F(n) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 6061 6061 54 4 4 1992 3 50090 54 8 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) + ---- F(n + 1) 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, 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(4 n - 2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 547 4 5 279 2 3 G(n) = ---- F(n) F(n + 1) + 24 F(n) F(n + 1) - --- F(n) F(n + 1) 22 22 2 4 739 3 2 3 3 + 13 F(n) F(n + 1) + --- F(n) F(n + 1) - 33 F(n) F(n + 1) 22 45 257 251 2 124 5 - -- F(n) F(n + 1) - --- F(n + 1) + --- F(n + 1) + --- F(n + 1) 22 22 22 11 6 51 2 - 11 F(n + 1) + -- F(n) - 7/22 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, 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 ----- \ 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) 4 335 2 2 3 G(n) = 1/12 F(n) F(n + 1) + --- F(n) F(n + 1) - 3 F(n) F(n + 1) 12 3 2 3 4 41 4 + 25/6 F(n) F(n + 1) - 105/2 F(n) F(n + 1) - -- F(n) F(n + 1) 12 3 2 3 5 - 25/6 F(n) - 381/4 F(n) F(n + 1) + 1/6 F(n + 1) + 7/6 F(n) 6 2 5 4 3 + 95 F(n) F(n + 1) - 425/2 F(n) F(n + 1) + 485/2 F(n) 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, 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(4 n - 2 j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 559 63 2 6 G(n) = ---- F(n) + --- F(n + 1) - 1575/2 F(n) F(n + 1) 319 638 85509 2 2 7 103185 3 + ----- F(n) F(n + 1) + 325 F(n) F(n + 1) - ------ F(n) F(n + 1) 638 319 4 4 2869 3 2 13292 3 + 1375/2 F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 4371 4 63 2834 4 241 4 + ---- F(n) - --- + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 638 638 319 638 471 2 3 + --- 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, 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(4 n - 2 j) F(n + 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 7 6 7 G(n) = -1/3 F(n) (24 F(n) + 46 F(n) F(n + 1) - 8 F(n + 1) 2 5 5 2 4 3 + 12 F(n) F(n + 1) + 734 F(n) F(n + 1) - 660 F(n) F(n + 1) 6 2 + 37 F(n) F(n + 1) - 185 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, 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(4 n - 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 5 447 2 G(n) = -65 F(n + 1) - 150 F(n) F(n + 1) - --- F(n) F(n + 1) 22 3 3 6 39 2 1439 2 + 160 F(n) F(n + 1) + 65 F(n + 1) - -- F(n) - ---- F(n + 1) 11 22 1439 3 3 223 6 + ---- F(n + 1) + 6/11 F(n) + --- F(n) F(n + 1) + 150 F(n) F(n + 1) 22 11 67 2 2 4 2 5 + -- F(n) F(n + 1) - 30 F(n) F(n + 1) + 30 F(n) F(n + 1) 22 3 4 - 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, 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 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 6 2927 5111 2 2 G(n) = 3475/4 F(n + 1) + 73/6 F(n + 1) + ---- + ---- F(n) F(n + 1) 12 12 3 2 2 4 2 - 3223/6 F(n) F(n + 1) - 37/3 F(n + 1) - 15 F(n) F(n + 1) - 7/4 F(n) 4 2 6 3 5 - 1423/6 F(n) - 75/2 F(n) F(n + 1) - 2325 F(n) F(n + 1) 7 8 5 + 5425/2 F(n) F(n + 1) - 2225/2 F(n + 1) + 61/6 F(n) F(n + 1) 3 3 5 + 175/6 F(n) F(n + 1) - 145/6 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, 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(4 n - 2 j) F(2 n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 233373 2 3 2 4 G(n) = 14525/2 F(n) F(n + 1) + ------ F(n) F(n + 1) + 45/2 F(n) F(n + 1) 319 3 6 5311 5 - 11325/2 F(n) F(n + 1) - ---- F(n) F(n + 1) + 120 F(n) F(n + 1) 319 1865 2 16779 2 6 6969 + ---- F(n) + ----- F(n + 1) - 105/2 F(n + 1) - ---- F(n) 638 319 638 30731 9 2 7 + ----- F(n + 1) - 5825/2 F(n + 1) - 625 F(n) F(n + 1) 319 147547 3 2 785951 4 898325 5 - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) + ------ F(n + 1) 319 638 319 3 3 - 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, 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(4 n - 2 j) F(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 3 4 19 7 G(n) = -775 F(n) F(n + 1) + 725 F(n) F(n + 1) - -- + 325 F(n + 1) 22 8 129 3 7159 3 2801 3 - 325 F(n + 1) + --- F(n) - ---- F(n + 1) - ---- F(n) F(n + 1) 22 22 22 7 404 2 775 2 2 + 775 F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 11 22 2 5 3 5 1381 2 - 50 F(n) F(n + 1) - 725 F(n) F(n + 1) + ---- F(n) F(n + 1) 11 3589 4 57 3 2 6 + ---- F(n + 1) - -- F(n) F(n + 1) + 50 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, 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 ----- \ G(n) = ) F(j) F(4 n - 2 j) F(3 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 5 9 131 9 G(n) = 409/4 F(n) F(n + 1) + 5/6 F(n) F(n + 1) + 1/6 F(n + 1) - --- F(n) 12 3 6 4 5 - 1/6 F(n + 1) + 1/4 F(n) F(n + 1) + 19/2 F(n) F(n + 1) 5 2 6 6 3 + 75/2 F(n) F(n + 1) - 16/3 F(n) F(n + 1) + 935/3 F(n) F(n + 1) 7 2 8 8 - 1457/4 F(n) F(n + 1) - 75/2 F(n) F(n + 1) - 3/4 F(n) F(n + 1) 2 2 3 3 6 - 29/4 F(n) F(n + 1) + 19/2 F(n) F(n + 1) - 67/4 F(n) F(n + 1) 4 3 35 7 - 95/3 F(n) F(n + 1) + -- 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, 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(4 n - 2 j) F(4 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 207 31 37 32935 5 3 5 G(n) = - --- + -- F(n) + -- F(n + 1) + ----- F(n + 1) + 3150 F(n) F(n + 1) 22 22 11 22 7 3 2 7 - 3650 F(n) F(n + 1) + 84 F(n) F(n + 1) - 25 F(n) F(n + 1) 1493 3 2 3 6 3 - ---- F(n) F(n + 1) - 3150 F(n) F(n + 1) + 632 F(n) F(n + 1) 22 7101 4 8 2 2 - ---- F(n) F(n + 1) + 3650 F(n) F(n + 1) - 250 F(n) F(n + 1) 11 2 3 4 9 + 246 F(n) F(n + 1) - 1491 F(n + 1) - 1500 F(n + 1) 2 6 8 + 25 F(n) F(n + 1) + 1500 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 ----- \ 2 G(n) = ) F(j) F(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 7 8 7 G(n) = -5/2 F(n) - 3/2 F(n) + 1/2 F(n + 1) - 1/2 F(n + 1) 23 6 41 2 2 2 - -- F(n) F(n + 1) + -- F(n) F(n + 1) - 42/5 F(n) F(n + 1) 10 10 6 2 4 3 4 4 - 108/5 F(n) F(n + 1) + 21/2 F(n) F(n + 1) - 32 F(n) F(n + 1) 5 2 6 3 - 37/2 F(n) F(n + 1) - 13/5 F(n) F(n + 1) - 16/5 F(n) F(n + 1) 7 7 5 3 + 16/5 F(n) F(n + 1) + 16 F(n) F(n + 1) + 56 F(n) F(n + 1) 2 + 14/5 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, 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 ----- \ G(n) = ) F(j) F(4 n - j) F(5 n - 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 260 3 6 5025 4 G(n) = -5/58 F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 29 638 31965 4 5 8135 5 4 15295 8 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 1276 319 1276 470 2 2 415 3 1350 3 2 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 5 3 625 2 7 2 6 + 5/58 F(n) F(n + 1) - --- F(n) F(n + 1) - 5/29 F(n) F(n + 1) 638 105 95 9 4 4 - ---- F(n + 1) + 5/638 + ---- F(n + 1) + 5/29 F(n) F(n + 1) 1276 1276 7 280 9 565 4 + 5/58 F(n) F(n + 1) + --- F(n) - --- 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(4 n - j) F(5 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 2 6 3 G(n) = 1/2 F(n) (1320 F(n) F(n + 1) - 20 F(n) F(n + 1) + 30 F(n) F(n + 1) 4 2 2 8 3 - 611 F(n + 1) - 92 F(n) F(n + 1) + 615 F(n + 1) + 247 F(n) F(n + 1) 7 - 1485 F(n) F(n + 1) - 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, 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 ----- \ G(n) = ) F(j) F(4 n - j) F(5 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 107 6 2 7201 9 12773 9 G(n) = ---- F(n + 1) - 3/11 F(n) + ---- F(n) - ----- F(n + 1) 22 902 902 109 2 5151 185 119 5 + --- F(n + 1) - ---- F(n) - --- F(n + 1) + --- F(n) F(n + 1) 22 902 902 11 148 3 3 16 2598 4 - --- F(n) F(n + 1) - -- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 11 451 26115 8 31 2 4 39156 6 3 + ----- F(n) F(n + 1) + -- F(n) F(n + 1) - ----- F(n) F(n + 1) 902 22 451 17862 7 2 5946 8 6438 5 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n + 1) 451 451 451 Proof: Both 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 ----- \ G(n) = ) F(j) F(4 n - j) F(5 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 3535 5 2347 52 G(n) = -266/3 F(n + 1) + 7/6 F(n) - ---- F(n + 1) - ---- F(n) + -- F(n + 1) 31 62 31 10480 9 2 2 5 + ----- F(n + 1) + 203/6 F(n) F(n + 1) - 50/3 F(n) F(n + 1) 93 4310 4 10045 8 2 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - 59/6 F(n) F(n + 1) 31 31 6330 2 3 3040 9 6 - ---- F(n) F(n + 1) + ---- F(n) - 210 F(n) F(n + 1) 31 93 6865 2 7 5510 8 3 4 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + 590/3 F(n) F(n + 1) 31 31 7 + 265/3 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, 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 ----- \ G(n) = ) F(j) F(4 n - j) F(5 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 103 3 G(n) = -27/2 F(n) F(n + 1) + --- F(n) F(n + 1) - 1/2 + 29/4 F(n) 10 9 5 8 5 3 + 1/2 F(n + 1) - 11/4 F(n) + 2 F(n) - 83/2 F(n) F(n + 1) 7 7 2 5 4 - 9/2 F(n) F(n + 1) + 195/4 F(n) F(n + 1) + 277/2 F(n) F(n + 1) 2 6 2 2 7 + 19 F(n) F(n + 1) + 3/5 F(n) F(n + 1) - 59/5 F(n) F(n + 1) 8 139 6 2 6 3 + 25/4 F(n) F(n + 1) + --- F(n) F(n + 1) - 75/2 F(n) F(n + 1) 10 4 5 8 - 85 F(n) F(n + 1) - 50 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, 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 ----- \ G(n) = ) F(j) F(4 n - j) F(6 n - 6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6875 9 34 3 16875 2 8 G(n) = ---- F(n) F(n + 1) + -- F(n) F(n + 1) - ----- F(n) F(n + 1) 14 21 14 7125 2 4 625 3 7 28125 4 6 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) + ----- F(n) F(n + 1) 28 14 28 34 2 2 2621 2 2617 6 37 4 - -- F(n) F(n + 1) + ---- F(n + 1) - ---- F(n + 1) - 1/21 - -- F(n) 21 84 84 42 6365 3 3 9005 5 377 5 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 42 21 42 13 2 + -- F(n) 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, 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 ----- \ G(n) = ) F(j) F(4 n - j) F(6 n - 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 288381 6 361 16109 6379 2 G(n) = ------ F(n + 1) + --- F(n) + ----- F(n + 1) - ----- F(n + 1) 26158 290 9570 13079 137914 10 279975 9 2968 9 2027 5 - ------ F(n + 1) + ------ F(n) F(n + 1) - ---- F(n) + ---- F(n + 1) 13079 13079 1595 1595 4011 5 72 4 31288 2 3 + ---- F(n) F(n + 1) + -- F(n) F(n + 1) + ----- F(n) F(n + 1) 1189 29 4785 26341 8 37471 7 2 1925265 6 4 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ------- F(n) F(n + 1) 4785 4785 26158 55655 14415 9 116535 8 2 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 13079 1189 26158 36358 8 631860 7 3 883 10 - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) - --- F(n) 4785 13079 638 14098 9 - ----- F(n + 1) 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, 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 ----- \ G(n) = ) F(j) F(4 n - j) F(6 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 9 5 G(n) = -1/2 F(n) (31 F(n) F(n + 1) + 47 F(n + 1) - 94 F(n + 1) + 47 F(n + 1) 6 3 5 4 3 2 + 547 F(n) F(n + 1) - 31 F(n) F(n + 1) - 265 F(n) F(n + 1) - 4 F(n) 2 7 4 - 250 F(n) F(n + 1) - 28 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, 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 ----- \ G(n) = ) F(j) F(4 n - j) F(6 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2100 8 2 57185 9 99 G(n) = ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - -- F(n) F(n + 1) 11 341 31 39948 5 670 6 1775 9 + ----- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 341 11 341 28 2 65 2 5 1325 3 4 + -- F(n) F(n + 1) + -- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 22 22 87960 7 3 113 2 19626 6 + ----- F(n) F(n + 1) - --- F(n) F(n + 1) - ----- F(n + 1) 341 11 341 3 567 3 1301 10 34779 10 - 9/22 F(n) + --- F(n + 1) + ---- F(n + 1) - ----- F(n) 22 682 682 565 7 16165 2 37889 2 - --- F(n + 1) + ----- F(n) + ----- F(n + 1) 22 341 682 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, 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 ----- \ G(n) = ) F(j) F(4 n - j) F(6 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 68084 9 19807 2 11646 10 8 G(n) = ------ F(n) F(n + 1) - ----- F(n) + ----- F(n) - 13/6 F(n) 615 410 205 20866 5 7 3 5 + ----- F(n) F(n + 1) + 175/6 F(n) F(n + 1) + 46 F(n) F(n + 1) 205 3 2 2 9119 2 8 - 175/6 F(n) F(n + 1) + 19 F(n) F(n + 1) + ---- F(n) F(n + 1) 41 2 6 14141 2 4 7 - 503/6 F(n) F(n + 1) + ----- F(n) F(n + 1) + 73/2 F(n) F(n + 1) 205 3 130172 7 3 42764 8 2 - 71/3 F(n) F(n + 1) - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 205 205 1487 83836 9 10 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - 1/3 + 1/3 F(n + 1) 205 615 Proof: Both 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 ----- \ G(n) = ) F(j) F(4 n - j) F(6 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 10 9 5 G(n) = 1/2 F(n + 1) - 1/2 F(n + 1) + 4174 F(n) F(n + 1) 9 3 3 3 2 - 4251 F(n) F(n + 1) + 851 F(n) F(n + 1) - 68 F(n) F(n + 1) 2 8 2 7 8 + 10437 F(n) F(n + 1) - 2349/2 F(n) F(n + 1) + 977/2 F(n) F(n + 1) 4 6 2 3 4 - 8687 F(n) F(n + 1) + 379/2 F(n) F(n + 1) + 43/2 F(n) F(n + 1) 4 2 4 5 3 6 - 849/2 F(n) F(n + 1) + 1062 F(n) F(n + 1) - 51 F(n) F(n + 1) 2 3 7 2 4 - 7 F(n) - 374 F(n) F(n + 1) - 3551/2 F(n) F(n + 1) 4 + 77 F(n) F(n + 1) - 485 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, 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(4 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 2 G(n) = - 1/2 + 1/2 F(n + 1) - 1/2 F(n) + F(n) F(n + 1) - 13/2 F(n) F(n + 1) 2 3 3 3 2 4 + 5 F(n) F(n + 1) + 7 F(n) F(n + 1) - 6 F(n) F(n + 1) - 2 F(n) 4 5 + 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, 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 ----- \ G(n) = ) F(j) F(4 n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 2 2 3 G(n) = -1/3 F(n) F(n + 1) - 10/3 F(n) F(n + 1) + 10/3 F(n) F(n + 1) 5 2 2 4 - 95/6 F(n) F(n + 1) - 1/3 + 1/3 F(n + 1) + 19/6 F(n) - 5/6 F(n) 2 4 3 3 4 2 + 13/6 F(n) F(n + 1) - 17/3 F(n) F(n + 1) + 52/3 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, 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 ----- \ G(n) = ) F(j) F(4 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 261 2 2 2 G(n) = -8/11 F(n) F(n + 1) + --- F(n) F(n + 1) + 2 F(n) F(n + 1) 11 26 3 3 2 5 - -- F(n) F(n + 1) - 1/11 F(n + 1) + 1/11 - 165 F(n) F(n + 1) 11 4 3 6 107 3 + 315/2 F(n) F(n + 1) + 70 F(n) F(n + 1) - --- F(n) 22 754 2 3 4 17 4 - --- F(n) F(n + 1) - 25/2 F(n) 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, 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 ----- \ G(n) = ) F(j) F(4 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 5 3 6 2 G(n) = -16 F(n) F(n + 1) + 125/4 F(n) F(n + 1) + 785/8 F(n) F(n + 1) 7 8 8 4 - 475/4 F(n) F(n + 1) - 79/2 F(n) + 5/8 F(n + 1) + 5/8 - 5/4 F(n + 1) 4 + 359/8 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, 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 ----- \ G(n) = ) F(j) F(4 n - j) F(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 215 2 2 385785 2 3 135 3 G(n) = --- F(n) F(n + 1) + ------ F(n) F(n + 1) - --- F(n) F(n + 1) 638 638 319 77930 3 2 54705 4 450790 4 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 319 638 319 82585 8 3240 2215595 2 7 + ----- F(n) F(n + 1) - ---- F(n) - ------- F(n) F(n + 1) 58 319 638 18120 3 6 3 86110 4 5 + ----- F(n) F(n + 1) - 5/29 F(n) F(n + 1) + ----- F(n) F(n + 1) 319 29 95 4 9 + 5/638 + --- F(n) - 5/638 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, 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(4 n - j) F(6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 274541 9 22332 5 10 G(n) = ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + 1/21 F(n + 1) 210 35 4 77997 2 10 2 2 - 5/42 F(n) - ----- F(n) - 1/21 - -- F(n) F(n + 1) 140 21 2 4 1027919 2 8 10 3 - 6625/7 F(n) F(n + 1) - ------- F(n) F(n + 1) + -- F(n) F(n + 1) 420 21 361283 3 3 240377 8 2 295318 9 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 105 60 105 140599 240781 10 - ------ F(n) F(n + 1) + ------ F(n) 210 420 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, 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(4 n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 6 2 G(n) = 1/2 F(n) - 11 F(n + 1) + 3/2 F(n) - 19/2 F(n + 1) + 23/2 F(n + 1) 5 3 3 + 9 F(n + 1) - 33 F(n) F(n + 1) - 5/2 F(n) F(n + 1) 4 5 2 3 - 37/2 F(n) F(n + 1) + 24 F(n) F(n + 1) - 29/2 F(n) F(n + 1) 2 4 3 2 + 13 F(n) F(n + 1) + 59/2 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, 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(4 n - j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 7 3 2 G(n) = 9/2 F(n + 1) - 485/2 F(n + 1) - 29/6 F(n) - 191/2 F(n) F(n + 1) 4 3 2 2 - 29/3 F(n) F(n + 1) + 91/6 F(n) F(n + 1) + 85/3 F(n) F(n + 1) 6 3 4 2 5 + 580 F(n) F(n + 1) - 1075/2 F(n) F(n + 1) + 30 F(n) F(n + 1) 2 3 3 - 41/6 F(n) F(n + 1) + 1457/6 F(n + 1) + 5/6 F(n) - 29/6 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, 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(4 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 207 4 2 3 19 4 G(n) = --- F(n) F(n + 1) - 3/11 F(n) F(n + 1) + -- F(n) F(n + 1) - 1/11 22 22 43 177 4 7 - -- F(n) + 1/11 F(n + 1) + --- F(n) + 325 F(n) F(n + 1) 22 22 3557 3 197 3 2 474 3 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 11 22 11 2 6 4 4 2955 2 2 - 1575/2 F(n) F(n + 1) + 1375/2 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, 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 ----- \ G(n) = ) F(j) F(4 n - j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 3 8 G(n) = 5510 F(n) F(n + 1) + 4308 F(n) F(n + 1) + 8900 F(n) F(n + 1) 2 7 3 6 - 525 F(n) F(n + 1) - 7200 F(n) F(n + 1) - 10 F(n) + 3600 F(n + 1) 9 3 2 4 - 3600 F(n + 1) - 7472 F(n) F(n + 1) - 3511 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(4 n - j) F(n + 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2554 6 160 5 4 4211 5 283 9 G(n) = ---- F(n + 1) - --- F(n) F(n + 1) + ---- F(n + 1) - ---- F(n + 1) 4785 29 9570 1914 1097 5 10491 6 1429 10657 4 5 + ---- F(n) + ----- F(n) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 9570 638 9570 1914 132677 4 6 555782 5 11087 5 5 + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 3828 4785 638 2012 6 3 64 6 4 1194 7 2 - ---- F(n) F(n + 1) - -- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 11 319 113533 7 3 4997 8 49 10 - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n + 1) 319 1914 348 2557 2 1700947 8 2 2671 907 - ---- F(n + 1) + ------- F(n) F(n + 1) - ---- F(n) - ---- F(n + 1) 6380 3828 4785 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, 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(4 n - j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 2 4 2 5 G(n) = -55/4 F(n) F(n + 1) - 23/2 F(n) F(n + 1) + 39/4 F(n) F(n + 1) 5 2 6 4 2 - 61/2 F(n) F(n + 1) + 75/2 F(n) F(n + 1) - 37/2 F(n) F(n + 1) 5 3 3 6 6 + 13 F(n) F(n + 1) + 18 F(n) F(n + 1) - 4 F(n) + 1/2 F(n + 1) 2 3 3 7 7 - F(n + 1) + 3/4 F(n + 1) - 17 F(n) + 17 F(n) - 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, 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(4 n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 6 2 5 G(n) = 23/3 F(n) - 2 F(n) + 1/3 - 1/3 F(n + 1) + 1/3 F(n) F(n + 1) 6 2 5 3 4 4 + 1097/6 F(n) F(n + 1) - 731/6 F(n) F(n + 1) - 136/3 F(n) F(n + 1) 5 2 6 3 + 19/3 F(n) F(n + 1) + 47/6 F(n) F(n + 1) - 154/3 F(n) F(n + 1) 4 2 3 3 2 4 - 59/6 F(n) F(n + 1) + 26/3 F(n) F(n + 1) - 31/6 F(n) F(n + 1) 3 5 + 131/6 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, 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(4 n - j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 9 2 7 G(n) = 120 F(n) F(n + 1) - 5825/2 F(n + 1) - 625 F(n) F(n + 1) 5292 3 2 26733 4 141 2281 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - --- F(n) + ---- F(n + 1) 11 22 11 22 30896 5 8087 2 3 2 4 + ----- F(n + 1) + ---- F(n) F(n + 1) + 45/2 F(n) F(n + 1) 11 11 3 6 181 8 - 11325/2 F(n) F(n + 1) - --- F(n) F(n + 1) + 14525/2 F(n) F(n + 1) 11 6 31 2 1157 2 3 3 - 105/2 F(n + 1) + -- F(n) + ---- F(n + 1) - 130 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, 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(4 n - j) F(2 n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 153408 6 2693 9 290934 5 G(n) = ------ F(n + 1) + ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 775 62 775 12443 278933 2 243857 2 268657 10 + ----- F(n) F(n + 1) - ------ F(n + 1) - ------ F(n) + ------ F(n) 1550 1550 1550 1550 27883 10 82426 9 285307 7 3 - ----- F(n + 1) + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 1550 155 310 8 2 + 1393/2 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, 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(4 n - j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 8 2 5 G(n) = -36 + 1/2 F(n + 1) - 145/4 F(n + 1) - 29/2 F(n) F(n + 1) 3 4 5 2 4 4 + 24 F(n) F(n + 1) + 52 F(n) F(n + 1) + 73/2 F(n) + 291/4 F(n + 1) 3 3 3 3 5 + 11/2 F(n) - F(n + 1) + 133/2 F(n) F(n + 1) + 50 F(n) F(n + 1) 4 3 4 4 6 - 71/2 F(n) F(n + 1) - 425/4 F(n) F(n + 1) - 37 F(n) F(n + 1) 2 6 5 3 + 625/4 F(n) F(n + 1) - 395/2 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, 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(4 n - j) F(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 7 5 2 G(n) = -1/3 F(n + 1) + 17/6 F(n) + 223/6 F(n) F(n + 1) 3 2 9 9 4 3 - 27/5 F(n) F(n + 1) - 77/6 F(n) + 1/3 F(n + 1) - 92/3 F(n) F(n + 1) 6 6 3 7 2 - 35/6 F(n) F(n + 1) + 973/3 F(n) F(n + 1) - 366 F(n) F(n + 1) 447 8 2 3 8 - --- F(n) F(n + 1) + 11 F(n) F(n + 1) - 3/2 F(n) F(n + 1) 10 6 3 6 4 + 1/2 F(n) F(n + 1) - 58/5 F(n) F(n + 1) + 546/5 F(n) F(n + 1) 3 4 2 + 1/3 F(n) F(n + 1) - 41/6 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(4 n - j) F(3 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1120 2 457 2 3 105 3 G(n) = ----- F(n) F(n + 1) + --- F(n) + 1/11 F(n + 1) - --- F(n) 11 22 22 1986 65636 5 6 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + 100 F(n) F(n + 1) 11 11 67652 9 326 2 28573 2 4 + ----- F(n) F(n + 1) + --- F(n) F(n + 1) + ----- F(n) F(n + 1) 11 11 11 2 5 336323 2 8 16098 3 3 - 225 F(n) F(n + 1) - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 22 11 3 4 19405 4 2 4 3 - 125/2 F(n) F(n + 1) + ----- F(n) F(n + 1) + 525/2 F(n) F(n + 1) 22 132824 4 6 12923 3 7 10 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - 1/11 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, 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(4 n - j) F(4 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 9 8 4 5 G(n) = 1/2 F(n + 1) - 1/2 F(n + 1) + 399/2 F(n) F(n + 1) 5 4 3 6 4 - 1301/2 F(n) F(n + 1) + 1003/2 F(n) F(n + 1) + 10 F(n) F(n + 1) 8 2 2 2 3 - 3/2 F(n) F(n + 1) - 481/2 F(n) F(n + 1) + 239 F(n) F(n + 1) 2 7 3 2 4 5 - 449/2 F(n) F(n + 1) - 64 F(n) F(n + 1) - 19/2 F(n) - 1/2 F(n) 4 4 5 3 3 5 - 397/2 F(n) F(n + 1) + 651 F(n) F(n + 1) - 502 F(n) F(n + 1) 2 6 7 3 + 447/2 F(n) F(n + 1) + 2 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, 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(4 n - j) F(4 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2384 2 6 5 3 G(n) = ----- F(n + 1) + 125/6 F(n) + 476/5 F(n) F(n + 1) 75 3 4 20336 8 2 + 64/5 F(n) F(n + 1) - 28/5 F(n + 1) + ----- F(n) F(n + 1) 75 28513 7 3 27299 9 37701 6 4 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 75 150 50 6 2 2477 5 4 4 - 62 F(n) F(n + 1) - ---- F(n) F(n + 1) - 40 F(n) F(n + 1) 25 30281 5 5 8 1762 413 10 - ----- F(n) F(n + 1) - 21/2 F(n) + ---- F(n) F(n + 1) + --- F(n + 1) 75 75 75 8 1996 6 - 2/5 F(n + 1) + 17/3 + ---- F(n + 1) 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, 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(4 n - j) F(5 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 9 2 4 G(n) = 1/4 F(n + 1) - 1/2 F(n + 1) - 980 F(n) F(n + 1) + 31 F(n) F(n + 1) 8 5 4 6 + 130 F(n) F(n + 1) - 85 F(n) F(n + 1) + 1/4 F(n + 1) 5 5 4 2 7 + 133/2 F(n) F(n + 1) - 215 F(n) F(n + 1) + 15 F(n) F(n + 1) 4 2 2 3 2 8 + 205 F(n) F(n + 1) - 35 F(n) F(n + 1) - 53/4 F(n) F(n + 1) 6 9 10 7 3 + 483 F(n) + 15 F(n) - 482 F(n) + 346 F(n) F(n + 1) 7 2 6 3 8 2 + 640 F(n) F(n + 1) - 480 F(n) F(n + 1) + 1435/4 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, 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(5 n - 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6125 4525 3025 3 6 G(n) = ---- F(n) F(n + 1) + ---- F(n) - ---- F(n) F(n + 1) 3781 7562 3781 3025 4 4525 2 408850 5 + ---- F(n) F(n + 1) - ---- F(n) + ------ F(n) F(n + 1) 3781 7562 3781 1711975 4 6 3025 4 5 40425 4 2 - ------- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 7562 7562 3781 3025 4 34475 3 7 3025 3 2 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 7562 3781 3781 86900 3 3 422400 9 353525 2 4 + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 3781 3781 7562 3025 2 3 3025 2 7 1028175 2 8 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ------- F(n) F(n + 1) 7562 7562 3781 3025 8 3025 10 3025 9 + ---- F(n) F(n + 1) + ---- F(n + 1) - ---- F(n + 1) 3781 7562 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, 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(5 n - 5 j) F(5 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 4 2 3 G(n) = 5/3 F(n + 1) F(n) (-2 F(n) + 7 F(n + 1) F(n) - 6 F(n + 1) F(n) 3 2 4 5 + 3 F(n + 1) F(n) - 4 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, 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(5 n - 5 j) F(5 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 130 2 55 10 455 10 G(n) = ---- F(n) F(n + 1) - --- F(n + 1) - ---- F(n) 551 551 1102 37435 5 5 730 2 455 7 55 3 + ----- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) + --- F(n + 1) 4408 551 1102 551 865 5 455 6 2475 9 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 2204 1102 4408 455 2 5 3850 4 6 15165 5 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 551 551 2204 455 6 21575 7 3 13100 8 2 + --- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 551 1102 551 16025 9 1875 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 4408 4408 Proof: Both 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(5 n - 5 j) F(5 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 10 95 4 85 6 155 5 G(n) = -5/42 F(n + 1) + 5/42 - -- F(n) + -- F(n) - --- F(n) F(n + 1) 84 84 21 6 4 20 7 25 + 1875/7 F(n) F(n + 1) + -- F(n) F(n + 1) + -- F(n) F(n + 1) 21 42 10 2 6 950 2 8 1275 3 3 + -- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 21 21 14 4 2 50 4 4 20675 4 6 + 175/6 F(n) F(n + 1) - -- F(n) F(n + 1) - ----- F(n) F(n + 1) 21 84 1865 5 5 50 6 2 3575 2 4 + ---- F(n) F(n + 1) + -- F(n) F(n + 1) - ---- F(n) F(n + 1) 42 21 84 2 2 - 10/7 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, 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(5 n - 5 j) F(5 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 46455 4 2 516375 4 6 G(n) = -5/638 F(n + 1) - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 638 319 20750 5 5 8465 248855 5 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) 319 638 319 23000 9 1295 2 3 17645 2 4 - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 29 638 58 611750 2 8 46445 3 3 105 4 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 29 40 4 75 2 75 5 + 5/638 F(n + 1) + --- F(n) F(n + 1) - -- F(n) + -- F(n) 319 58 58 80 3 2 + -- 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, 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 ----- \ 2 G(n) = ) F(j) F(5 n - 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 295 25 G(n) = 5/638 F(n + 1) - 5/638 F(n + 1) + --- F(n) - --- F(n) F(n + 1) 638 319 135 2 75 2 3 125 3 2 - --- F(n) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 638 58 700 4 25 5 - --- 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, 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(5 n - 5 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 2 G(n) = -5/84 (-2 F(n + 1) + F(n)) (10 F(n) - 15 F(n) F(n + 1) - 11 F(n) 2 2 2 + 15 F(n + 1) F(n) + F(n) F(n + 1) + 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, 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 ----- \ G(n) = ) F(j) F(5 n - 5 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 55 55 1375 965 3 G(n) = - --- + --- F(n + 1) + ---- F(n) - ---- F(n) F(n + 1) 551 551 1102 1102 610 2 2 825 2 3 975 3 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 551 551 1102 4675 3 2 715 4 275 4 275 5 + ---- F(n) F(n + 1) - ---- F(n) - --- F(n) F(n + 1) - --- F(n) 1102 1102 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, 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(5 n - 5 j) F(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 43450 83050 2 83875 5 169125 6 G(n) = ------ F(n + 1) - ----- F(n + 1) + ----- F(n + 1) + ------ F(n + 1) 3781 3781 7562 7562 16775 16775 2 203500 5 49500 2 3 + ----- F(n) - ----- F(n) - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 7562 7562 3781 3781 10625 2 4 251625 3 2 347125 3 3 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 3781 7562 7562 3625 9625 4 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 398 398 Proof: Both 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(5 n - 5 j) F(6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 38575 6 6035 2 3 4 G(n) = ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - 1700/9 F(n) F(n + 1) 198 198 805 2 650 2 3 1825 2 5 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 99 99 99 3 2 5 400 4 + 50/3 F(n) F(n + 1) + 50/9 F(n + 1) - --- F(n) F(n + 1) 33 8150 7 3 8170 3 190 - ---- F(n + 1) - 10/9 F(n) + ---- F(n + 1) - --- F(n + 1) + 10/9 F(n) 99 99 33 Proof: Both 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(5 n - 5 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 75 2 260 2 2510 2 245 2 G(n) = ---- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n + 1) - --- F(n) 638 319 319 638 5025 6 5625 5 6375 3 3 + ---- F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 319 319 760 2425 2 4 245 3 3 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) - 5/638 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, 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(5 n - 5 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 215 2 5 205 2 2 G(n) = --- F(n) - 5/14 F(n) F(n + 1) - --- F(n) F(n + 1) 84 84 25 2 4 205 3 200 4 2 + -- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 84 84 21 115 5 95 4 6 55 6 - --- F(n) F(n + 1) - -- F(n) + 5/42 F(n + 1) - -- F(n) - 5/42 12 84 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, 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(5 n - 5 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 55 55 2 1625 2 590 5 1035 6 G(n) = ---- F(n + 1) + --- F(n + 1) - ---- F(n) + --- F(n) + ---- F(n) 551 551 551 551 551 1175 4 1585 5 695 2 3 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 1102 1102 551 800 2 4 555 3 2 1390 4 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 551 551 551 10025 4 2 11075 5 - ----- F(n) F(n + 1) + ----- F(n) 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, 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(5 n - 5 j) F(n + 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1793275 6 127925 2 227275 2 4 G(n) = -------- F(n) F(n + 1) - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 7562 7562 7562 136575 2 5 721150 3 4 76980 + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 7562 3781 3781 726225 7 490975 6 339465 2 + ------ F(n + 1) + ------ F(n + 1) + ------ F(n) F(n + 1) 7562 7562 7562 13570 3 361600 3 1132775 5 + ----- F(n) - ------ F(n + 1) - ------- F(n) F(n + 1) 3781 3781 7562 604475 3 3 13570 2 13000 2 + ------ F(n) F(n + 1) - ----- F(n) - ----- F(n + 1) 3781 3781 199 Proof: Both 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(5 n - 5 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 515 6 13410 3 515 3 G(n) = - --- - 100 F(n) F(n + 1) - ----- F(n + 1) + --- F(n) 638 319 638 3085 2 255 4 10495 2 - ---- F(n) F(n + 1) + --- F(n + 1) + ----- F(n) F(n + 1) 638 319 638 925 7 565 3 2075 3 4 + --- F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 22 319 22 125 2 5 860 3 115 2 2 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 22 319 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, 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(5 n - 5 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 6 1025 2 G(n) = 5/84 F(n) F(n + 1) - 5/28 F(n) F(n + 1) + ---- F(n) F(n + 1) 84 2 3 215 3 2 1415 3 4 - 25/7 F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 42 84 445 4 40 5 3515 5 2 - --- F(n) F(n + 1) + -- F(n) - ---- F(n) F(n + 1) 84 21 84 345 4 3 3 40 3 + --- F(n) F(n + 1) + 5/42 F(n + 1) - 5/42 F(n + 1) - -- F(n) 14 21 335 2 5 - --- 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, 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(5 n - 5 j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 12485 2 2 5 2925 3 3 G(n) = ------ F(n) F(n + 1) + 175/2 F(n) F(n + 1) + ---- F(n) F(n + 1) 1102 551 3 4 4 3 4595 + 25 F(n) F(n + 1) - 100 F(n) F(n + 1) + ---- F(n) F(n + 1) 551 6 18925 2 4 26475 4 2 - 75/2 F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 1102 1102 55 2 2005 2 55 3 2005 3 - --- F(n + 1) - ---- F(n) + --- F(n + 1) + ---- F(n) 551 1102 551 1102 21560 2 10575 5 + ----- F(n) F(n + 1) - ----- F(n) 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, 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(5 n - 5 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 380 50 2 6 5200 7 26250 3 G(n) = - --- + -- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 11 11 319 7915 4 940 2 2 2955 2 3 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 29 638 1755 5 4275 8 380 3515 + ---- F(n + 1) + ---- F(n + 1) + --- F(n) - ---- F(n + 1) 319 22 319 638 4500 3 5 4540 3 2 61605 4 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n + 1) 11 319 319 3455 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, 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(5 n - 5 j) F(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 85 8 13 2 2 1387 3 G(n) = -- F(n) - -- F(n) F(n + 1) + ---- F(n) F(n + 1) + 5/42 28 42 84 1675 2 4 785 5 3 4 2 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) - 75/8 F(n) F(n + 1) 168 14 1625 3 3 471 6 2 445 7 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 84 14 28 5 1387 7 3245 2 6 + 5/42 F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 84 84 2 285 2 1385 6 - 5/42 F(n + 1) + --- F(n) - ---- F(n) 56 168 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, 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(5 n - 5 j) F(3 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 84875 4 328545 4 3185 3 1375 3 G(n) = ------ F(n) - ------ F(n + 1) + ---- F(n) + ---- F(n + 1) 1102 2204 1102 2204 385785 5 3 3 5 55 7 + ------ F(n) F(n + 1) - 150 F(n) F(n + 1) - -- F(n + 1) 1102 76 165385 8 689175 4 4 33275 2 5 + ------ F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 2204 2204 2204 661625 2 6 70565 3 8635 2 - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 2204 551 551 39975 3 4 40845 43875 4 3 - ----- F(n) F(n + 1) + ----- - ----- F(n) F(n + 1) 1102 551 2204 51105 5 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, 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 ----- \ G(n) = ) F(j) F(5 n - 5 j) F(4 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 9205 5 5975 2 4 18575 6 G(n) = ----- F(n + 1) - ---- F(n) F(n + 1) + ----- F(n + 1) 11 638 638 1200 2 7 1085 3 2 37975 3 6 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 11 11 22 3120 46425 8 2010 2 3 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 22 11 1275 6015 7915 4 1275 2 + ---- F(n) - ---- F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) 638 319 22 638 9290 2 18825 9 21425 3 3 - ---- F(n + 1) + ----- F(n + 1) + ----- F(n) F(n + 1) 319 22 319 21775 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, 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(5 n - 5 j) F(4 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 9 3 365 G(n) = -5/12 F(n) F(n + 1) + 5/42 F(n + 1) - 5/42 F(n + 1) - --- F(n) 56 925 5 425 7 265 4 195 8 + --- F(n) + --- F(n) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 28 84 28 56 605 2 165 4 4 3 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - 75/2 F(n) F(n + 1) 84 14 1455 4 5 1325 5 2 1075 6 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 14 28 84 6 3 5305 9 25 6 + 720/7 F(n) F(n + 1) - ---- F(n) + -- F(n) F(n + 1) 168 42 10695 5 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, 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(5 n - 5 j) F(5 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 185 6 185 3 17 2 23605 10 G(n) = ---- F(n) + --- F(n) + -- F(n) F(n + 1) + ----- F(n + 1) 58 58 22 29 20325 4 2 6575 4 3 8330 5 - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 22 638 319 6141 5 2 1088435 5 5 11545 6 + ---- F(n) F(n + 1) + ------- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 638 188925 2 8 37600 3 3 21375 3 7 - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 58 29 11 1272900 4 6 1038405 6 565 3 4 + ------- F(n) F(n + 1) - ------- F(n + 1) + --- F(n) F(n + 1) 319 638 638 8950 2 167 3 339 7 + ---- F(n + 1) + --- F(n + 1) - --- F(n + 1) 11 319 638 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, 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 ----- \ 2 G(n) = ) F(j) F(5 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 72680 10 48 3 81231 2 309715 10 G(n) = ----- F(n) - --- F(n) - ----- F(n) - ------ F(n + 1) 9367 551 9367 18734 289543 2 195 2 441 3 - ------ F(n + 1) + ---- F(n) F(n + 1) - ---- F(n + 1) 18734 1102 1102 492925 2 8 606755 6 591545 5 - ------ F(n) F(n + 1) + ------ F(n + 1) - ------ F(n) F(n + 1) 9367 18734 9367 51489 204775 2 4 193 2 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 9367 9367 1102 549875 9 284400 9 + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 9367 9367 Proof: Both 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(5 n - 4 j) F(5 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 19 2 2 8 15 4 5 G(n) = -- F(n) - 8439/7 F(n) F(n + 1) + -- F(n) - 3378/7 F(n) F(n + 1) 28 28 631 3 7 10 + --- F(n) F(n + 1) + 3/14 F(n + 1) - 3/14 - 64/7 F(n) F(n + 1) 14 1381 4 2 1369 3 3 3 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + 2/7 F(n) F(n + 1) 28 14 2 4 2 2 6869 9 + 1436/7 F(n) F(n + 1) - 9/7 F(n) F(n + 1) + ---- F(n) F(n + 1) 14 28131 4 6 + ----- 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, 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(5 n - 4 j) F(5 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 63 10 8523 5 20115 4 6 G(n) = ---- F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 638 638 475 9 289 8 289 3 6 + --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 638 319 319 189 3 2 29 4 5 42259 5 5 + --- F(n) F(n + 1) - -- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 22 58 226 5 4 29 2 7 157 2 3 - --- F(n) F(n + 1) + -- F(n) F(n + 1) - --- F(n) F(n + 1) 319 22 319 767 2 509 5 159 4 211083 2 4 - --- F(n) - --- F(n) - --- F(n) F(n + 1) - ------ F(n) F(n + 1) 638 638 638 638 84251 3 3 29467 4 2 63 9 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + --- F(n + 1) 638 638 638 506475 3 7 210865 2 8 - ------ 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, 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(5 n - 4 j) F(5 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 2 7 G(n) = 1/2 F(n) (6 F(n) - 2697 F(n + 1) + 250 F(n) F(n + 1) - 53 F(n + 1) 9 4 8 + 2750 F(n + 1) + 1137 F(n) F(n + 1) - 6750 F(n) F(n + 1) 2 3 3 2 3 6 - 543 F(n) F(n + 1) + 275 F(n) F(n + 1) + 5625 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, 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 ----- \ 2 G(n) = ) F(j) F(5 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 G(n) = 1/6 F(n + 1) - 1/6 F(n + 1) - 1/12 F(n) + 1/4 F(n) F(n + 1) 11 2 2 3 3 - -- F(n) F(n + 1) + 5/2 F(n) F(n + 1) - 1/12 F(n) 12 3 2 35 4 5 - 15/4 F(n) F(n + 1) + -- F(n) F(n + 1) - 5/6 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, 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(5 n - 4 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 63 63 343 53 3 35 2 2 G(n) = - --- + --- F(n + 1) + --- F(n) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 638 638 319 58 58 945 2 3 33 3 1085 3 2 + --- F(n) F(n + 1) - -- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 58 319 15 4 2485 4 315 5 + -- F(n) - ---- F(n) F(n + 1) - --- 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, 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 ----- \ G(n) = ) F(j) F(5 n - 4 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 933 2 2373 5 441 441 2 G(n) = ---- F(n) + ---- F(n) - ---- F(n + 1) + ---- F(n + 1) 1102 1102 1102 1102 84 4 27 5 6951 2 3 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 551 1102 1102 4897 2 4 4746 3 2 2640 3 3 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 1102 551 551 7119 4 6587 4 2 267 5 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 1102 1102 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, 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(5 n - 4 j) F(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1635 6 15 4 9005 5 2 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 77 154 308 240 2 3 3 2 5345 3 4 - --- F(n) F(n + 1) + 30/7 F(n) F(n + 1) - ---- F(n) F(n + 1) 77 308 45 4 2510 4 3 85 7 15 5 - -- F(n) F(n + 1) + ---- F(n) F(n + 1) - -- F(n) + -- F(n) 14 77 14 14 855 2 765 6 15 3 15 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + -- F(n + 1) - -- F(n + 1) 308 308 77 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, 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 ----- \ G(n) = ) F(j) F(5 n - 4 j) F(6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6874 5 36260 2 3 32480 4 G(n) = ---- F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 3781 3781 3781 201884 3 84805 4 4 205961 5 3 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 18905 15124 18905 232801 6 2 193455 7 69365 91218 4 + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) + ----- - ----- F(n + 1) 7562 3781 15124 18905 79123 8 10487 8 20622 5 13748 6496 + ----- F(n) + ----- F(n + 1) - ----- F(n) + ----- F(n) - ---- F(n + 1) 15124 75620 3781 3781 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, 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(5 n - 4 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 25 2 2 2 4 G(n) = -1/6 F(n) F(n + 1) - -- F(n) F(n + 1) + 5/12 F(n) F(n + 1) 12 6 4 37 2 2 - 2 F(n) - 1/6 + 1/12 F(n) + -- F(n) + 1/6 F(n + 1) 12 13 3 4 2 125 5 + -- F(n) F(n + 1) + 10 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, 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(5 n - 4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 390 3 2 1030 3 3 625 4 G(n) = ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 638 2280 4 2 2125 5 415 5 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 427 2 3 2 4 727 4 - --- F(n) F(n + 1) + 5/638 F(n) F(n + 1) - --- F(n) F(n + 1) 638 638 529 5 63 63 2 747 2 - --- F(n) - --- F(n + 1) + --- F(n + 1) - --- 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, 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(5 n - 4 j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 756 3 441 7 441 6 32593 2 G(n) = ---- F(n) + ---- F(n + 1) - ---- F(n + 1) - ----- F(n) F(n + 1) 551 1102 1102 1102 21313 5 39371 2 4 1902 4 2 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 1102 1102 29 44373 4 3 11297 31453 6 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 551 551 1102 9179 2 65981 2 5 16064 3 3 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 1102 1102 551 12899 3 4 1999 2 - ----- F(n) F(n + 1) - ---- F(n) 551 551 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, 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(5 n - 4 j) F(n + 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 1150 4 2 41953 4 4 G(n) = 45/7 F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 77 3080 106947 6 2 48907 7 15 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + -- F(n) F(n + 1) 770 385 77 655 5 625 2 84099 4 120513 8 3489 + --- F(n) F(n + 1) + --- F(n + 1) + ----- F(n) - ------ F(n) - ---- 77 154 1540 3080 616 1247 4 587 8 20 6 655 6 + ---- F(n + 1) + ---- F(n + 1) - -- F(n) - --- F(n + 1) 220 3080 11 154 4658 3 - ---- F(n) F(n + 1) 385 Proof: Both 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(5 n - 4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 457 5 2 283 3 7 G(n) = ---- F(n) F(n + 1) - 4/3 F(n) + 3/4 F(n + 1) - --- F(n + 1) - 2 F(n) 30 60 11 5 5 293 7 2 - -- F(n + 1) + 4/3 F(n) + --- F(n + 1) + 3/20 F(n) F(n + 1) 12 60 6 4 3 101 6 - 61/6 F(n) F(n + 1) + 62/3 F(n) F(n + 1) + --- F(n) F(n + 1) 12 4 61 2 3 + 13/4 F(n) F(n + 1) - -- 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, 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(5 n - 4 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4308 2 1717 559 3 398 2 G(n) = ---- F(n + 1) - ---- F(n) F(n + 1) + --- F(n) + --- F(n) 319 319 319 319 946 3 3 1811 5 24899 2 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 29 58 638 7165 2 6 2 5 - ---- F(n) F(n + 1) - 475/2 F(n) F(n + 1) - 25/2 F(n) F(n + 1) 638 57 2 4 3 4 789 6 7 + -- F(n) F(n + 1) + 225 F(n) F(n + 1) - --- F(n + 1) + 100 F(n + 1) 58 58 63737 3 - ----- 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, 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(5 n - 4 j) F(2 n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 441 7 68434 6 303491 2 5 G(n) = ----- F(n + 1) - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 1102 551 1102 358591 4 3 2445 4 441 8 - ------ F(n) F(n + 1) + ---- F(n) + ---- F(n + 1) 1102 1102 1102 40884 3 4 58713 3 5 418911 4 4 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 551 1102 1102 40523 2 87752 3 174133 7 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 1102 551 1102 62119 2 2 200538 2 6 6371 3 + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) + ---- F(n) 1102 551 1102 16343 3 69190 2 - ----- 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, 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(5 n - 4 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 2 6 347 5 G(n) = 79/6 F(n + 1) - 25/3 F(n) F(n + 1) - --- F(n) F(n + 1) 12 335 3 3 3 37 3 - --- F(n) F(n + 1) + 143/4 F(n) F(n + 1) + -- - 1265/6 F(n) F(n + 1) 12 12 2 2 8 5965 4 - 40/3 F(n + 1) - 1/12 F(n) - 500 F(n + 1) + ---- F(n + 1) 12 7 2 2 149 2 4 + 3650/3 F(n) F(n + 1) + 250/3 F(n) F(n + 1) - --- F(n) F(n + 1) 12 3 5 - 1050 F(n) F(n + 1) + 11/4 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, 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 ----- \ G(n) = ) F(j) F(5 n - 4 j) F(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 591 4 10369 4 31 8 1645 8 1325 7 G(n) = --- F(n + 1) - ----- F(n) + ---- F(n + 1) + ---- F(n) - ---- F(n) 638 638 1276 116 638 1149 7 10888 5 2 21919 4 4 - ---- F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 1595 1595 1276 3467 6 14419 6 2 16043 7 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 638 319 319 6918 3 5 1605 4 3 479 2 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 3190 4099 3 4 1983 3 1087 - ---- F(n) F(n + 1) + ---- F(n + 1) - ---- 638 3190 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, 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(5 n - 4 j) F(4 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 142 2 7 853 3 2 G(n) = 65/6 F(n) - 1/6 F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 15 30 5 2 626 7 2 8 + 105/4 F(n) F(n + 1) - --- F(n) F(n + 1) + 28 F(n) F(n + 1) 15 8 2 5 5 4 - 1/12 F(n) F(n + 1) - 31/6 F(n) F(n + 1) - 459/4 F(n) F(n + 1) 6 3 37 2 3 11 2 + 160 F(n) F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 15 12 2 305 6 7 + 1/4 F(n) F(n + 1) - --- F(n) F(n + 1) - 43/4 F(n) + 1/6 F(n + 1) 12 61 9 - -- 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, 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(5 n - 4 j) F(4 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 34567 2 2 248883 2 3 303113 8 G(n) = ----- F(n) F(n + 1) - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 638 638 319 649931 4 5 18805 2 6 1467337 2 7 - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ------- F(n) F(n + 1) 319 58 638 11279 3 46318 3 2 24051 3 6 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 638 319 638 28647 4 985 3 5 86205 3 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 58 638 16885 4 4 604379 4 3860 7 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 58 638 29 63 9 63 1092 4 2983 + --- F(n + 1) - --- + ---- F(n) + ---- F(n) 638 638 319 638 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, 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(5 n - 4 j) F(5 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 7 3 2 4 G(n) = 13775/3 F(n) F(n + 1) + 555/4 F(n) F(n + 1) + 13025/6 F(n) F(n + 1) 527 2 2 10055 7 3 3 - --- F(n) F(n + 1) - ----- F(n) F(n + 1) - 4520/3 F(n) F(n + 1) 12 12 6 4195 8 2 - 4961/2 F(n + 1) - 1/6 + ---- F(n + 1) + 2065/6 F(n + 1) 12 4195 4 4 6 3 5 - ---- F(n + 1) + 32795/3 F(n) F(n + 1) + 2290/3 F(n) F(n + 1) 12 2 8 2 6 3 - 25205/2 F(n) F(n + 1) - 145/4 F(n) F(n + 1) + 8 F(n) F(n + 1) 5 10 - 10603/3 F(n) F(n + 1) + 49/6 F(n) - 305/6 F(n) F(n + 1) 10 + 4273/2 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, 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 ----- \ 2 G(n) = ) F(j) F(5 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 43130 3 3 23000 3 7 425 4 G(n) = ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - --- F(n) F(n + 1) 319 29 319 13230 4 2 12 4 5 885 4 6 - ----- F(n) F(n + 1) - -- F(n) F(n + 1) - --- F(n) F(n + 1) 319 11 29 3576 5 24 5 4 232506 5 5 + ---- F(n) F(n + 1) + -- F(n) F(n + 1) + ------ F(n) F(n + 1) 319 11 319 12 6 3 560 105110 2 8 + -- F(n) F(n + 1) - --- F(n) F(n + 1) + ------ F(n) F(n + 1) 11 319 319 128 10 188 5 412 543 6 128 5 + --- F(n + 1) - --- F(n) + --- F(n) - --- F(n) - --- F(n + 1) 319 319 319 319 319 356 3 2 24 3 6 107045 2 4 - --- F(n) F(n + 1) - -- F(n) F(n + 1) - ------ F(n) F(n + 1) 319 11 319 12 2 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, 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(5 n - 3 j) F(5 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 2 2 G(n) = -2173/2 F(n) F(n + 1) + 2 F(n) + 265/2 F(n + 1) 3 7 9 2 4 - 5500 F(n) F(n + 1) + 7000 F(n) F(n + 1) + 709 F(n) F(n + 1) 2 8 3 3 - 47/2 F(n) F(n + 1) - 1125/2 F(n) F(n + 1) - 1077/2 F(n) F(n + 1) 10 6 - 5625/2 F(n + 1) + 2680 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, 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(5 n - 3 j) F(5 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 7 19 2 7265 3 3 G(n) = 1/11 F(n) + 1/11 F(n + 1) - -- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 22 34 6 10 59 3 4 15 9 - -- F(n) - 1/11 F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 11 11 44 65 5 581 5 73 5 2 + -- F(n) F(n + 1) + --- F(n) F(n + 1) - -- F(n) F(n + 1) 44 22 22 38 6 8950 6 4 13885 3 7 + -- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 11 11 44 2425 4 2 61 4 3 16125 4 6 - ---- F(n) F(n + 1) + -- F(n) F(n + 1) + ----- F(n) F(n + 1) 22 22 22 6 6877 5 5 - 9/22 F(n) F(n + 1) + ---- F(n) 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, 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(5 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 17 3 15 2 3 G(n) = - 1/11 + 1/11 F(n + 1) + -- F(n) - F(n) F(n + 1) + -- F(n) F(n + 1) 11 11 3 65 3 2 70 4 5 - F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) - 5/11 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, 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(5 n - 3 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 128 128 2 106 2 744 5 G(n) = ---- F(n + 1) + --- F(n + 1) - --- F(n) + --- F(n) 319 319 319 319 104 4 5 2128 2 3 + --- F(n) F(n + 1) + 4/319 F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 1906 2 4 2976 3 2 2875 3 3 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 2232 4 2241 4 2 634 5 - ---- 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, 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(5 n - 3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 195 4 783 2 195 2 3 G(n) = ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 28 28 28 2 5 165 3 2 3 117 3 + 30 F(n) F(n + 1) + --- F(n) F(n + 1) + 1699/7 F(n + 1) - --- F(n) 14 28 2669 2 33 5 3 4 - ---- F(n) F(n + 1) + -- F(n) - 1075/2 F(n) F(n + 1) 28 28 6 99 5 7 + 580 F(n) F(n + 1) + -- F(n + 1) - 15/4 F(n + 1) - 485/2 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, 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(5 n - 3 j) F(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 165 5 4445 1665 4 19325 4 G(n) = --- F(n + 1) - ---- + ---- F(n + 1) + ----- F(n) 551 2204 551 2204 4 4 2435 8 55 5 1020 - 100 F(n) F(n + 1) - ---- F(n + 1) + --- F(n) - ---- F(n) 2204 551 551 110 4550 3 2 41635 3 - --- F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 551 551 1102 5100 4 290875 6 2 67275 3 5 + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 551 2204 1102 34855 5 3 - ----- 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, 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(5 n - 3 j) F(6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4648 6 3 8 G(n) = -181/9 F(n + 1) + ---- F(n) F(n + 1) - 37/3 F(n) F(n + 1) 15 443 2 3 5 5 694 4 - --- F(n) F(n + 1) + 181/9 F(n + 1) + 368/9 F(n) - --- F(n) F(n + 1) 15 15 272 8 9 5 4 + --- F(n) F(n + 1) - 440/9 F(n) - 1978/9 F(n) F(n + 1) 45 Proof: Both 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(5 n - 3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 27 4 71 4 2 2 3 G(n) = -- F(n) F(n + 1) - -- F(n) F(n + 1) - 5/22 F(n) F(n + 1) 22 11 24 2 4 26 3 2 73 3 3 - -- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) F(n + 1) 11 11 11 25 4 37 5 42 5 - -- F(n) F(n + 1) + -- F(n) F(n + 1) + -- F(n) F(n + 1) 22 22 11 2 5 19 6 - 1/11 F(n + 1) + 1/11 F(n + 1) - 3/22 F(n) - -- 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, 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(5 n - 3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6255 5 10870 2 4 20565 4 2 G(n) = ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 319 128 2 6415 6 - --- F(n + 1) + ---- F(n) F(n + 1) + 20 F(n) F(n + 1) 319 319 929 2 2 5 9615 3 3 + --- F(n) F(n + 1) - 35 F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 3 4 4 3 166 3 1123 2 - 30 F(n) F(n + 1) + 65 F(n) F(n + 1) + --- F(n) - ---- F(n) 319 319 6432 2 128 3 - ---- 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, 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 ----- \ G(n) = ) F(j) F(5 n - 3 j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 6 1763 3 6 G(n) = -75/2 F(n) F(n + 1) - ---- F(n) F(n + 1) + 225/7 F(n + 1) 28 2 8 2225 3 3 - 7/4 F(n) - 2225/2 F(n + 1) + ---- F(n) F(n + 1) 28 2075 5 2645 2 2 3 5 - ---- F(n) F(n + 1) + 27/4 + ---- F(n) F(n + 1) - 2325 F(n) F(n + 1) 28 14 425 2 4 3 281 - --- F(n) F(n + 1) - 3321/7 F(n) F(n + 1) + --- F(n) F(n + 1) 28 28 7 453 2 30967 4 + 5425/2 F(n) F(n + 1) - --- 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, 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 ----- \ G(n) = ) F(j) F(5 n - 3 j) F(n + 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 19091 5 6022 5 167 6 1614 6 G(n) = ----- F(n + 1) - ---- F(n) + --- F(n + 1) + ---- F(n) 4408 551 290 551 679945 4 24635 4 2 292437 5 4 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 8816 1102 2204 5766455 6 3 24233 2623 2 + ------- F(n) F(n + 1) - ----- F(n + 1) - ---- F(n + 1) 8816 8816 5510 14829 9 22412 5 14951 - ----- F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 8816 2755 5510 1286345 3 6 4 5 1116083 3 2 + ------- F(n) F(n + 1) - 625 F(n) F(n + 1) - ------- F(n) F(n + 1) 4408 4408 17261 3 3 - ----- 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, 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 ----- \ G(n) = ) F(j) F(5 n - 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 43 3 4 2 G(n) = -1/11 F(n + 1) + 1/11 F(n + 1) + -- F(n) + 7/2 F(n) F(n + 1) 22 148 4 3 317 5 2 127 6 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 11 22 11 20 2 6 127 5 + -- F(n) F(n + 1) + 1/22 F(n) - --- F(n) F(n + 1) 11 44 233 3 4 65 5 + --- F(n) F(n + 1) + 9/44 F(n) F(n + 1) - -- F(n) F(n + 1) 22 44 21 2 5 181 3 3 + -- 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, 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 ----- \ G(n) = ) F(j) F(5 n - 3 j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 16166 3 4 4 4 216 6 G(n) = ------ F(n) F(n + 1) + 325 F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 7 11765 2 11399 2 2 + 125 F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 2 6 272 4 1337 3 - 275 F(n) F(n + 1) - --- F(n) - ---- F(n) F(n + 1) 319 319 40091 5 2 1867 3 128 3 3 5 + ----- F(n) F(n + 1) + ---- F(n) - --- F(n + 1) - 75 F(n) F(n + 1) 319 319 319 23493 4 3 39983 3 7543 2 5 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 319 128 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, 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 ----- \ G(n) = ) F(j) F(5 n - 3 j) F(2 n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 153 3041 41 3 2 5 G(n) = ---- F(n) + ---- F(n + 1) + -- F(n) - 25 F(n) F(n + 1) 14 28 14 9 133465 5 1759 2 - 4875 F(n + 1) + ------ F(n + 1) + ---- F(n) F(n + 1) 28 28 3 2 2 7 - 4007/7 F(n) F(n + 1) - 1325/2 F(n) F(n + 1) 3 6 57677 4 29637 2 3 - 19575/2 F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 28 28 517 2 8 6 - --- F(n) F(n + 1) + 24075/2 F(n) F(n + 1) - 775/2 F(n) F(n + 1) 28 3 4 3 7 + 725/2 F(n) F(n + 1) - 1139/7 F(n + 1) + 325/2 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, 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 ----- \ G(n) = ) F(j) F(5 n - 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 80 6 133 2 8 549 7 G(n) = --- F(n) F(n + 1) + --- F(n) F(n + 1) + 1/11 F(n + 1) - --- F(n) 11 22 110 7 389 3 4 529 3 1923 2 5 - 1/11 F(n + 1) - --- F(n) F(n + 1) + --- F(n) + ---- F(n) F(n + 1) 22 110 110 31 8 614 3 5 7 - -- F(n) + --- F(n) F(n + 1) + 29/2 F(n) F(n + 1) 11 11 643 2 541 7 1154 2 2 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 110 110 55 673 2 6 5743 6 2 173 3 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 22 110 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, 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 ----- \ G(n) = ) F(j) F(5 n - 3 j) F(3 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1833441 8 2475892 7 2 G(n) = -------- F(n) F(n + 1) - ------- F(n) F(n + 1) 1595 1595 1501051 7 2184421 2 3 81274 2 2 + ------- F(n) F(n + 1) + ------- F(n) F(n + 1) - ----- F(n) F(n + 1) 6699 1595 6699 1554132 8 1339175 7 464351 3 + ------- F(n) F(n + 1) + ------- F(n) F(n + 1) - ------ F(n) F(n + 1) 1595 6699 2233 407867 5 116596 4 553684 2 6 + ------ F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 1595 319 2233 827296 9 297505 354608 9 234715 8 356798 - ------ F(n + 1) - ------ - ------ F(n) + ------ F(n) + ------ F(n) 1595 6699 1595 6699 1595 420069 26787 4 267710 8 + ------ F(n + 1) + ----- F(n + 1) - ------ F(n + 1) 1595 319 6699 Proof: Both 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 ----- \ G(n) = ) F(j) F(5 n - 3 j) F(4 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 1117 4 4 5 G(n) = 1895/4 F(n + 1) + ---- + 1715/4 F(n) F(n + 1) - 4075/2 F(n) F(n + 1) 22 3 2 8 930 2 2 + 2185/2 F(n) F(n + 1) - 950 F(n) F(n + 1) + --- F(n) F(n + 1) 11 2 3 2 7 2393 3 - 3455/4 F(n) F(n + 1) + 2300 F(n) F(n + 1) - ---- F(n) F(n + 1) 22 3 6 20841 105 2 6 + 75/2 F(n) F(n + 1) - ----- F(n + 1) + --- F(n) - 575/2 F(n) F(n + 1) 44 22 3 5 4 4 8 + 125/2 F(n) F(n + 1) + 575/2 F(n) F(n + 1) + 50 F(n + 1) 2219 4 556 4 - ---- 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, 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(5 n - 3 j) F(5 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 41021 3 6 48127 2 4 4397 2 7 G(n) = ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 308 66 42 1936441 2 8 15325 2 3 514799 9 + ------- F(n) F(n + 1) - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 990 154 495 28327 5 20770 8 1551923 8 2 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ------- F(n) F(n + 1) 55 231 495 1125329 9 73979 2 9 17893 9 - ------- F(n) F(n + 1) + ----- F(n) - 1/11 F(n + 1) + ----- F(n) 495 165 924 1867 4 11161 8 1661 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) 44 462 84 123694 3 3 95 3 2 521737 + ------ F(n) F(n + 1) + --- F(n) F(n + 1) + ------ F(n) F(n + 1) 45 231 990 150463 10 10 - ------ F(n) + 1/11 F(n + 1) 330 Proof: Both 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 ----- \ 2 G(n) = ) F(j) F(5 n - 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 278 3 78500 3 7 G(n) = -178 F(n + 1) + --- F(n + 1) + ----- F(n) F(n + 1) 11 11 15350 5 670 6 99875 9 + ----- F(n) F(n + 1) + --- F(n) F(n + 1) - ----- F(n) F(n + 1) 11 11 11 67 2 7850 3 3 20325 2 4 + -- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 22 11 22 1325 3 4 353 565 7 3 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n + 1) - 9/22 F(n) 22 11 22 79 2 40125 10 65 2 5 16125 2 8 - -- F(n) + ----- F(n + 1) + -- F(n) F(n + 1) + ----- F(n) F(n + 1) 22 11 22 22 215 2 76325 6 - --- F(n) F(n + 1) - ----- 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, 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(5 n - 2 j) F(5 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 2 4 G(n) = 17 F(n) F(n + 1) + 527/3 F(n) F(n + 1) + 8 F(n) - 2 F(n) 8 4 4 4 6 - 1/3 F(n + 1) - 1226/3 F(n) F(n + 1) + 28126/3 F(n) F(n + 1) 5 9 - 251/3 F(n) F(n + 1) - 13502/3 F(n) F(n + 1) + 13748/3 F(n) F(n + 1) 2 2 2 4 2 6 - 65 F(n) F(n + 1) + 5749/3 F(n) F(n + 1) + 1226/3 F(n) F(n + 1) 2 8 3 3 3 5 - 33751/3 F(n) F(n + 1) - 916 F(n) F(n + 1) + 41 F(n) F(n + 1) 3 7 4 2 7 + 1252/3 F(n) F(n + 1) + 1376/3 F(n) F(n + 1) - 523/3 F(n) F(n + 1) 10 + 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, 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(5 n - 2 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 219 251 2 105 5 6 21 G(n) = ---- F(n + 1) + --- F(n + 1) + --- F(n + 1) - 11 F(n + 1) + -- F(n) 22 22 11 11 2 465 4 5 + 1/11 F(n) - --- F(n) F(n + 1) + 24 F(n) F(n + 1) 22 255 2 3 2 4 315 3 2 - --- F(n) F(n + 1) + 13 F(n) F(n + 1) + --- F(n) F(n + 1) 22 11 3 3 21 - 33 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, 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(5 n - 2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 2 3 G(n) = -1/4 F(n) F(n + 1) + 1/12 F(n) F(n + 1) - 8/3 F(n) F(n + 1) 4 341 6 3 4 - 11/4 F(n) F(n + 1) + --- F(n) F(n + 1) - 43/3 F(n) F(n + 1) 12 71 2 5 11 5 59 3 3 + -- F(n) F(n + 1) + -- F(n) - -- F(n) + 1/6 F(n + 1) - 1/6 F(n + 1) 12 12 12 3 2 289 4 3 5 2 + 11/3 F(n) F(n + 1) + --- F(n) F(n + 1) - 229/6 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, 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(5 n - 2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 3 5 3060 5 612 G(n) = 1700 F(n) F(n + 1) - 1375 F(n) F(n + 1) - ---- F(n + 1) - --- F(n) 319 319 6183 8914 3 12555 4 + ---- F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 638 29 638 4130 2 2 585 2 3 9180 3 2 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 29 58 319 1714 3 2 6 8 2526 - ---- F(n) F(n + 1) - 100 F(n) F(n + 1) - 1375/2 F(n + 1) + ---- 29 319 19705 4 + ----- 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, 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 ----- \ G(n) = ) F(j) F(5 n - 2 j) F(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 6 4 4 G(n) = 5/6 F(n) (87 F(n) F(n + 1) + 300 F(n) F(n + 1) - 300 F(n) F(n + 1) 5 3 4 3 5 + 870 F(n) F(n + 1) - 10 F(n) - 630 F(n) F(n + 1) 2 2 4 - 317 F(n) F(n + 1) + 2 F(n + 1) - 2) Proof: Both 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(5 n - 2 j) F(6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1001417 6 26871 5 17034363 10 2115 9 G(n) = ------- F(n) + ----- F(n) - -------- F(n) - ---- F(n) 6380 12122 121220 1276 13719 4 9809477 5 36921 + ----- F(n) F(n + 1) - ------- F(n) F(n + 1) - ----- F(n) F(n + 1) 12122 242440 96976 53309 9 13665 8 74307 5 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 96976 24244 48488 1561609 5 5 200493 5 4 - ------- F(n) F(n + 1) - ------ F(n) F(n + 1) 16720 24244 923793 4 6 53175 4 5 34539181 9 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - -------- F(n) F(n + 1) 24244 6061 96976 10551623 8 2 33690 7 2 39645 6 3 + -------- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 24244 6061 6061 54 10 54 9 24165 - ---- F(n + 1) + ---- F(n + 1) - ----- F(n) 6061 6061 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, 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(5 n - 2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 535 6 305 5 245 6 G(n) = ---- F(n) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 22 11 44 373 2 3 3 226 2 2 + --- F(n) F(n + 1) + 115/2 F(n) F(n + 1) + --- F(n) - 9/22 F(n + 1) 44 11 320 2 4 2 5 1345 4 3 - --- F(n) F(n + 1) - 5/44 F(n) F(n + 1) + ---- F(n) F(n + 1) 11 44 5 2 2 - 30 F(n) F(n + 1) + 3/22 F(n) F(n + 1) - 2/11 F(n) F(n + 1) 3 7 + 9/22 F(n + 1) - 5/22 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, 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(5 n - 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 3 G(n) = 1/6 - 31/8 F(n) F(n + 1) - 945/8 F(n) F(n + 1) 6 2 7 3 + 731/4 F(n) F(n + 1) + 31/8 F(n) F(n + 1) - 205/4 F(n) F(n + 1) 3 5 175 4 2 3 3 + 145/8 F(n) F(n + 1) - --- F(n) F(n + 1) + 10 F(n) F(n + 1) 12 4 2 4 4 55 5 + 31/4 F(n) - 1/6 F(n + 1) - 75/2 F(n) F(n + 1) + -- F(n) F(n + 1) 12 5 23 6 + 13/6 F(n) F(n + 1) - 2 F(n) F(n + 1) - -- 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, 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(5 n - 2 j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 21324 2 3 14585 2 4 G(n) = 14525/2 F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 29 638 3 6 5212 41205 3 3 - 11325/2 F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 37865 5 2 7 13944 3 2 + ----- F(n) F(n + 1) - 625 F(n) F(n + 1) - ----- F(n) F(n + 1) 319 29 70501 4 1805 2 16629 2 8185 - ----- F(n) F(n + 1) + ---- F(n) + ----- F(n + 1) - ---- F(n) 58 638 319 638 33029 9 81457 5 33195 6 + ----- F(n + 1) - 5825/2 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, 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(5 n - 2 j) F(n + 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 7 5 3 G(n) = 1/10 F(n) (3545 F(n) F(n + 1) - 1530 - 12306 F(n) F(n + 1) 7 8 4 8 + 5225 F(n) F(n + 1) - 1588 F(n + 1) + 3153 F(n + 1) + 1690 F(n) 3 6 2 - 3589 F(n) F(n + 1) + 5400 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, 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(5 n - 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1373 3 7 403 2 G(n) = ----- F(n) F(n + 1) + 775 F(n) F(n + 1) - --- F(n) F(n + 1) 11 11 403 2 2 2 5 7 + --- F(n) F(n + 1) - 50 F(n) F(n + 1) + 7/22 + 325 F(n + 1) 11 6 3 4 2 6 - 775 F(n) F(n + 1) + 725 F(n) F(n + 1) + 50 F(n) F(n + 1) 69 3 2759 2 8 - -- F(n) F(n + 1) + ---- F(n) F(n + 1) - 325 F(n + 1) 11 22 3 5 7159 3 125 3 3576 4 - 725 F(n) F(n + 1) - ---- 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, 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(5 n - 2 j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 9 9 7 65 6 G(n) = 1/6 F(n + 1) - 77/6 F(n) + 17/6 F(n) - -- F(n) F(n + 1) 12 17 2 5 7445 6 3 7 2 + -- F(n) F(n + 1) + ---- F(n) F(n + 1) - 1457/4 F(n) F(n + 1) 12 24 985 8 3 6 2 3 - --- F(n) F(n + 1) - 77/4 F(n) F(n + 1) + 61/6 F(n) F(n + 1) 24 2 4 355 4 5 + 1/4 F(n) F(n + 1) + 845/8 F(n) F(n + 1) + --- F(n) F(n + 1) 24 5 2 385 4 3 8 + 75/2 F(n) F(n + 1) - --- F(n) F(n + 1) - 3/4 F(n) F(n + 1) 12 2 3 - 43/6 F(n) 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, 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(5 n - 2 j) F(2 n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 197874 9 1255616 7 3 29266 2 G(n) = ------- F(n) F(n + 1) - ------- F(n) F(n + 1) - ----- F(n + 1) 4495 4495 8525 114589 2 62849 3 2664806 2 7 + ------ F(n) F(n + 1) + ----- F(n) - ------- F(n) - 235/8 F(n) 2552 2552 247225 7757 8 2 2 5 935223 10 + ---- F(n) F(n + 1) + 385/4 F(n) F(n + 1) - ------ F(n + 1) 29 44950 542548 6 1417117 10 2535406 + ------ F(n + 1) + ------- F(n) - ------- F(n) F(n + 1) 22475 44950 247225 1292 9 6 127121 5 + ---- F(n) F(n + 1) - 875/8 F(n) F(n + 1) + ------ F(n) F(n + 1) 31 22475 36979 2 39749 3 7 - ----- F(n) F(n + 1) - ----- F(n + 1) + 125/4 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, 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(5 n - 2 j) F(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 5255 2 3 G(n) = -6/11 F(n) + 150 F(n) F(n + 1) + ---- F(n) F(n + 1) 22 33559 2 7 727 3 7141 8 - ----- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 22 11 11 2649 2 2 7152 4 1641 3 6 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 11 11 4 33009 4 5 712 3 2 + 21/2 F(n) F(n + 1) + ----- F(n) F(n + 1) - --- F(n) F(n + 1) - 9/22 22 11 199 4 7 14297 3 - --- F(n) - 650 F(n) F(n + 1) + ----- F(n) F(n + 1) 22 22 4 4 2 6 9 - 1500 F(n) F(n + 1) + 1525 F(n) F(n + 1) + 9/22 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, 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(5 n - 2 j) F(3 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 391 3 7 G(n) = -1/6 F(n + 1) + --- F(n) F(n + 1) + 2/3 F(n) F(n + 1) 12 9 2 4 + 30524/3 F(n) F(n + 1) + 83/4 F(n) - 19/4 F(n) - 20351/2 F(n) F(n + 1) 2 2 2 6 2 8 - 481/4 F(n) F(n + 1) + 112 F(n) F(n + 1) - 75038/3 F(n) F(n + 1) 3 3 3 5 3 7 - 12053 F(n) F(n + 1) - 752/3 F(n) F(n + 1) + 6227/6 F(n) F(n + 1) 4 2 4 4 + 63067/3 F(n) F(n + 1) - 199/2 F(n) F(n + 1) 4 6 119701 5 5 3 + 62138/3 F(n) F(n + 1) + ------ F(n) F(n + 1) + 976/3 F(n) F(n + 1) 12 188371 2 4 10 - ------ F(n) F(n + 1) + 1/6 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, 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 ----- \ G(n) = ) F(j) F(5 n - 2 j) F(4 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5161 3 2 193 33643 5 G(n) = ---- F(n) F(n + 1) - --- F(n) F(n + 1) - ----- F(n) F(n + 1) 11 11 11 33816 8 33816 9 27821 2 3 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 11 11 22 27683 2 4 163359 2 7 163359 2 8 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 22 22 22 5144 3 3 1091 3 6 1091 3 7 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 11 11 3199 4 3285 4 2 145209 4 5 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 22 22 22 145209 4 6 333 33669 4 19 2 + ------ F(n) F(n + 1) + --- F(n) + ----- F(n) F(n + 1) + -- F(n) 22 22 11 22 9 10 - 9/22 F(n + 1) + 9/22 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, 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 ----- \ 2 G(n) = ) F(j) F(5 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 10 4 2 5 2 G(n) = 1/2 F(n + 1) - 849/2 F(n) F(n + 1) + 4174 F(n) F(n + 1) - 8 F(n) 9 8 4 6 - 1/2 F(n + 1) + 977/2 F(n) F(n + 1) - 8687 F(n) F(n + 1) 4 5 2 3 9 + 1062 F(n) F(n + 1) + 389/2 F(n) F(n + 1) - 4251 F(n) F(n + 1) 4 4 3 7 - 487 F(n) F(n + 1) + 37/2 F(n) F(n + 1) - 374 F(n) F(n + 1) 3 6 3 3 3 2 - 51 F(n) F(n + 1) + 851 F(n) F(n + 1) - 69 F(n) F(n + 1) 2 8 2 7 2 4 + 10437 F(n) F(n + 1) - 2349/2 F(n) F(n + 1) - 3551/2 F(n) F(n + 1) + 77 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, 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 ----- \ 2 G(n) = ) F(j) F(5 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 3 3 4 G(n) = 16 F(n) F(n + 1) - 15 F(n) F(n + 1) - 12 F(n) F(n + 1) 4 2 5 4 5 + 6 F(n) F(n + 1) + F(n) F(n + 1) + 3/2 F(n) F(n + 1) - F(n) F(n + 1) 2 3 2 4 2 - 21/2 F(n) F(n + 1) + 11 F(n) F(n + 1) - 1/2 F(n + 1) + 1/2 F(n + 1) 2 5 - 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, 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(5 n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 7 2 G(n) = -52/3 F(n) F(n + 1) - 29/6 F(n) - 1/3 F(n + 1) + 11/6 F(n) F(n + 1) 2 4 2 3 + 101/6 F(n) F(n + 1) + 1/6 F(n) F(n + 1) - 16/3 F(n) F(n + 1) 2 5 3 2 4 + 119/3 F(n) F(n + 1) + 22/3 F(n) F(n + 1) - 11/2 F(n) F(n + 1) 5 2 6 5 - 105/2 F(n) F(n + 1) + 107/6 F(n) F(n + 1) + 11/6 F(n) 3 + 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, 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(5 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2019 6 2 245 2 2 159 2 6 G(n) = ---- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 44 44 44 5 3 41 3 31 5 - 15 F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) + 1/11 F(n + 1) 22 22 635 7 389 4 4 163 8 93 4 - --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) + -- F(n) F(n + 1) 22 44 22 22 21 4 36 2 3 62 3 2 - -- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 11 8 - 1/11 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, 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(5 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 7 3 G(n) = 1/2 F(n) (3400 F(n + 1) - 8250 F(n) F(n + 1) + 173 F(n) F(n + 1) 3 5 4 3 + 7200 F(n) F(n + 1) - 3382 F(n + 1) + 1420 F(n) F(n + 1) 2 2 - 543 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, 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(5 n - j) F(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 885 10 80 1211 2 4 2 7 G(n) = --- F(n) - --- F(n) - ---- F(n) F(n + 1) + 5/638 F(n) F(n + 1) 58 319 319 63 3 6 35 4 5 10 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - 5/638 F(n + 1) 638 638 9 9 35 7 2 + 5/638 F(n + 1) - 5/638 F(n) + --- F(n) F(n + 1) 319 250131 7 3 12 8 125815 8 2 - ------ F(n) F(n + 1) + --- F(n) F(n + 1) + ------ F(n) F(n + 1) 638 319 638 27375 9 3237 2 8 807 3 2 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 319 638 638 77361 3 3 86135 3 7 393 4 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + --- F(n) F(n + 1) 638 638 319 88500 4 6 405 + ----- 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, 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(5 n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 3 3 6 G(n) = -65 F(n + 1) + 131/2 F(n + 1) - 1/2 F(n) + 150 F(n) F(n + 1) 2 2 4 2 5 + 5 F(n) F(n + 1) - 35 F(n) F(n + 1) + 30 F(n) F(n + 1) 3 4 2 2 - 160 F(n) F(n + 1) + 33/2 F(n) F(n + 1) - 58 F(n + 1) - 5/2 F(n) 2 5 6 - 21 F(n) F(n + 1) - 130 F(n) F(n + 1) + 115/2 F(n + 1) 3 3 + 295/2 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, 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 ----- \ G(n) = ) F(j) F(5 n - j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 4 2 3 G(n) = 2225/2 F(n) F(n + 1) - 4/3 F(n) - 975/2 F(n) F(n + 1) + 1/3 2 3 5 2 4 4 - 1/3 F(n + 1) - 100 F(n) F(n + 1) + 25/2 F(n) F(n + 1) + 7 F(n) 2 6 3 3 3 - 1150 F(n) F(n + 1) - 52 F(n) F(n + 1) + 50/3 F(n) F(n + 1) 7 2 2 + 26/3 F(n) F(n + 1) + 975/2 F(n) F(n + 1) + 367/2 F(n) F(n + 1) 5 4 2 - 25/3 F(n) F(n + 1) - 175/6 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, 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(5 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 411 2 133 9 9 5 G(n) = ---- F(n) - --- F(n) - 1/11 F(n + 1) + 55/2 F(n) F(n + 1) 22 22 2 1453 8 49 8 + 1/11 F(n + 1) + ---- F(n) F(n + 1) + -- F(n) F(n + 1) 22 22 235 2 4 15 2 7 531 4 5 + --- F(n) F(n + 1) + -- F(n) F(n + 1) - --- F(n) F(n + 1) 11 11 22 29 1145 3 3 67 5 240 6 - -- F(n) F(n + 1) - ---- F(n) F(n + 1) - -- F(n) + --- F(n) 22 22 11 11 1097 7 2 855 6 3 2 3 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) - 10 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, 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 ----- \ G(n) = ) F(j) F(5 n - j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 8 4 4 8 G(n) = 1/10 F(n) (150 F(n) + 4580 F(n) F(n + 1) + 35 F(n + 1) 5 3 3 6 2 - 5980 F(n) F(n + 1) + 73 F(n) F(n + 1) + 1219 F(n) F(n + 1) 7 7 2 2 - 915 F(n) F(n + 1) - 313 F(n) F(n + 1) + 1151 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, 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(5 n - j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 21 3 8 4 4 7 G(n) = -- F(n + 1) + 13/4 F(n + 1) - 13/2 F(n + 1) + 7 F(n) + 13/2 F(n) 10 7 3 4 8 67 2 - 13/5 F(n + 1) + 65/2 F(n) F(n + 1) - 45/4 F(n) + -- F(n) F(n + 1) 10 373 5 2 4 3 4 4 + --- F(n) F(n + 1) + 15/4 - 63 F(n) F(n + 1) + 263/4 F(n) F(n + 1) 10 6 6 2 7 - 25 F(n) F(n + 1) + 51/2 F(n) F(n + 1) - 36 F(n) F(n + 1) 3 5 - 46 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, 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(5 n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 9 6 7 3 G(n) = -73/6 F(n) - 45 F(n) F(n + 1) - 13 F(n) + 97/6 F(n) 4 3 3 6 4 5 + 35/4 F(n) F(n + 1) - 23 F(n) F(n + 1) + 121/6 F(n) F(n + 1) 1241 4 125 2 3 8 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) - 113/3 F(n) F(n + 1) 12 12 7 2 6 3 5 - 725/2 F(n) F(n + 1) + 1825/6 F(n) F(n + 1) - 1/12 F(n + 1) 3 5 2 + 5/12 F(n + 1) - 7/12 F(n + 1) + 38 F(n) F(n + 1) 2 5 7 - 31/4 F(n) F(n + 1) + 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, 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(5 n - j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 50 3 215 2 10 G(n) = 1/11 F(n + 1) - -- F(n) + --- F(n) - 1/11 F(n + 1) 11 11 336323 2 8 1371 3 4 19303 4 2 - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 22 22 22 5777 4 3 132824 4 6 3911 + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 22 11 22 65668 5 1098 6 67652 9 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 11 11 11 324 2 28656 2 4 16117 3 3 + --- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 11 11 11 12923 3 7 2237 2 2476 2 5 + ----- 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, 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 ----- \ G(n) = ) F(j) F(5 n - j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 9 3 G(n) = -9 - 1500 F(n + 1) + 19/2 F(n + 1) + 83 F(n) F(n + 1) 2 6 2 7 2 3 + 25 F(n) F(n + 1) - 25 F(n) F(n + 1) + 250 F(n) F(n + 1) 2 2 8 7 - 501/2 F(n) F(n + 1) + 3650 F(n) F(n + 1) - 3650 F(n) F(n + 1) 3 3 6 3 5 + 1267/2 F(n) F(n + 1) - 3150 F(n) F(n + 1) + 3150 F(n) F(n + 1) 4 3 2 5 - 632 F(n) F(n + 1) - 84 F(n) F(n + 1) + 1491 F(n + 1) 4 8 - 2983/2 F(n + 1) + 1500 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, 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 ----- \ G(n) = ) F(j) F(5 n - j) F(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2291 9 4 4 3853 3 3 G(n) = ----- F(n) F(n + 1) - 100 F(n) F(n + 1) - ---- F(n) F(n + 1) 12 60 3 5771 2 8 2 6 - 95/3 F(n) F(n + 1) + ---- F(n) F(n + 1) - 65/3 F(n) F(n + 1) 15 1714 2 4 2 2 + ---- F(n) F(n + 1) + 40/3 F(n) F(n + 1) - 1/3 15 3 5 76987 7 3 7 + 250/3 F(n) F(n + 1) - ----- F(n) F(n + 1) + 170/3 F(n) F(n + 1) 60 10 5 9 + 126 F(n) + 757/4 F(n) F(n + 1) + 1837/6 F(n) F(n + 1) 8 2 2 8 10 + 1057/2 F(n) F(n + 1) - 213/2 F(n) - 25/6 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, 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 ----- \ G(n) = ) F(j) F(5 n - j) F(4 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 10 10 25 6 97 9 G(n) = 17/9 F(n + 1) - 1/2 F(n) - -- F(n + 1) + -- F(n) + 91/9 F(n) 18 18 5 3 2 376 3 3 + 13/9 F(n + 1) - 1489/9 F(n) F(n + 1) - --- F(n) F(n + 1) 15 3 6 3 7 4 + 1969/9 F(n) F(n + 1) - 133/5 F(n) F(n + 1) - 69 F(n) F(n + 1) 7469 4 2 8639 4 5 8665 6 4 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 45 18 18 7 2 7 3 649 8 + 1505/3 F(n) F(n + 1) - 1504/3 F(n) F(n + 1) - --- F(n) F(n + 1) 18 1871 8 2 9 - ---- F(n) F(n + 1) - 110/9 F(n) F(n + 1) - 43/9 F(n) F(n + 1) 30 35 9 - -- 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, 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(6 n - 6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 2 G(n) = -2/3 F(n + 1) (4 F(n) - 30 F(n + 1) F(n) - 6 F(n + 1) F(n) 2 3 5 + 32 F(n + 1) F(n) - 13 F(n + 1) + 13 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, 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 ----- \ G(n) = ) F(j) F(6 n - 6 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 97065 5 106380 3 3 85545 6 G(n) = ------ F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n + 1) 6061 6061 12122 7767 2 2358 2 2358 3 54 3 - ---- F(n + 1) - ---- F(n) + ---- F(n) - ---- F(n + 1) 1102 6061 6061 6061 17280 2 4 13599 1028 2 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 6061 6061 4606 2 - ---- 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, 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 ----- \ G(n) = ) F(j) F(6 n - 6 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 32 2 96 2 32 464 2 4 G(n) = --- F(n + 1) + -- F(n) - --- + --- F(n) F(n + 1) 231 77 231 231 256 3 3 544 5 176 4 2 - --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 77 77 21 256 4 32 5 584 2 2 - --- F(n) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 231 231 231 584 3 + --- F(n) F(n + 1) 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, 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 ----- \ G(n) = ) F(j) F(6 n - 6 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 378 378 2 4050 2 4050 5 G(n) = ----- F(n + 1) + ---- F(n + 1) - ---- F(n) + ---- F(n) 3781 3781 3781 3781 4208 4 5211 5 4418 2 3 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 3781 3781 3781 2079 2 4 3464 3 2 13284 3 3 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 7562 3781 3781 9364 4 53001 4 2 23571 5 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 3781 7562 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, 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 ----- \ G(n) = ) F(j) F(6 n - 6 j) F(6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 173760 3 4 11785920 2 5 G(n) = ------- F(n) F(n + 1) + -------- F(n) F(n + 1) 103679 103679 9880240 4 3 4651880 6 - ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 103679 103679 1749552 2 2108448 3616560 2 4 - ------- F(n) F(n + 1) + ------- F(n) F(n + 1) + ------- F(n) F(n + 1) 103679 103679 103679 6718320 4 2 372096 3 372096 2 - ------- F(n) F(n + 1) + ------ F(n) - ------ F(n) 103679 103679 103679 4630344 2 2067840 5 + ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 103679 103679 3100320 3 3 41472 2 41472 3 + ------- F(n) F(n + 1) - ------ F(n + 1) + ------ F(n + 1) 103679 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, 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 ----- \ G(n) = ) F(j) F(6 n - 6 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 G(n) = 1/6 F(n) F(n + 1) (-71 F(n + 1) + 14 F(n) - 48 F(n + 1) F(n) 2 3 5 4 + 188 F(n + 1) F(n) + 71 F(n + 1) - 154 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, 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 ----- \ G(n) = ) F(j) F(6 n - 6 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 508 2 2 4768 3 41317 4 3 G(n) = ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 551 6061 6061 5592 4 736 12081 7 4856 7 74061 5 2 - ---- F(n) + ---- - ----- F(n + 1) + ---- F(n) + ----- F(n) F(n + 1) 6061 6061 12122 6061 12122 14604 6 22947 3 4 1065 6 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 12122 319 790 4 12189 3 - ---- 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, 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 ----- \ G(n) = ) F(j) F(6 n - 6 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 40 5 1910 2 5 1940 3 4 G(n) = -- F(n) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 21 21 77 3560 4 3 32 3 40 3 32 + ---- F(n) F(n + 1) + --- F(n + 1) - -- F(n) - --- F(n + 1) 33 231 21 231 16 4 9613 2 4 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) - 16/3 F(n) F(n + 1) 231 231 9565 6 2806 2 288 2 3 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 231 231 77 3 2 + 16/3 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, 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 ----- \ G(n) = ) F(j) F(6 n - 6 j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 100166 5 2 144545 4 3 22581 4 2 G(n) = ------ F(n) F(n + 1) - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 3781 7562 3781 13915 6 44585 2 1959 2 4 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ---- F(n) F(n + 1) 7562 7562 3781 23889 2 5 18 3 3 17534 5 + ----- F(n) F(n + 1) - -- F(n) F(n + 1) + ----- F(n) F(n + 1) 7562 19 3781 5981 7 6874 7 2111 2 275 3 + ---- F(n + 1) + ---- F(n) + ---- F(n + 1) - --- F(n + 1) 7562 3781 3781 398 6874 6 131 6 - ---- F(n) - --- F(n + 1) 3781 199 Proof: Both 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 ----- \ G(n) = ) F(j) F(6 n - 6 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 4 3 3 G(n) = -5/2 F(n) F(n + 1) - 36 F(n) F(n + 1) - 45/2 + 40 F(n) F(n + 1) 4 3 5 5 3 8 + 45/2 F(n) + 50 F(n) F(n + 1) - 74 F(n) F(n + 1) - 9/2 F(n + 1) 4 + 27 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, 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 ----- \ G(n) = ) F(j) F(6 n - 6 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 18200 1308212 6 2 138511 4 81464 G(n) = ------ F(n + 1) - ------- F(n) F(n + 1) + ------ F(n + 1) + ----- 6061 30305 12122 18183 18146 5 7214 5 216440 7 + ----- F(n + 1) + ---- F(n) - ------ F(n) F(n + 1) 6061 6061 18183 148 3 2 149428 7 1918666 2 2 + --- F(n) F(n + 1) + ------ F(n) F(n + 1) + ------- F(n) F(n + 1) 19 4785 90915 25852 2 3 303566 3 38316 4 - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 30305 6061 103106 8 578137 8 - ------ F(n) - ------ F(n + 1) 18183 36366 Proof: Both 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 ----- \ G(n) = ) F(j) F(6 n - 6 j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2162 3 27400 5 2 6 G(n) = ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - 25/3 F(n) F(n + 1) 77 231 19265 2 2 3832 4600 2 4 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 231 231 231 3 5 48658 3 7 - 1050 F(n) F(n + 1) - ----- F(n) F(n + 1) + 3650/3 F(n) F(n + 1) 231 3944 2 104 2 38268 4 104 11800 6 - ---- F(n + 1) - --- F(n) + ----- F(n + 1) + --- + ----- F(n + 1) 77 33 77 33 231 9400 3 3 8 + ---- F(n) F(n + 1) - 500 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, 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(6 n - 6 j) F(2 n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 12266 2 446161 6 2 190862 6 G(n) = ----- F(n) F(n + 1) - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 18905 18905 18905 232391 4 4 36878 4 3 39604 5 2 - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 7562 3781 3781 5992 2 72487 7 37226 6 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 18905 18905 18905 42512 3 274003 2 2 54419 7 - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 18905 37810 3781 211654 5 3 378 8 10924 8 10924 7 + ------ F(n) F(n + 1) + ---- F(n + 1) - ----- F(n) + ----- F(n) 3781 3781 3781 3781 378 7 - ---- 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, 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(6 n - 6 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 3 4 G(n) = -1/6 F(n) F(n + 1) (-F(n + 1) - 53 F(n) + 50 F(n + 1) F(n) 2 5 6 7 5 2 - 268 F(n + 1) F(n) + 270 F(n + 1) F(n) + 9 F(n) - 8 F(n + 1) F(n) 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, 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(6 n - 6 j) F(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 12070 12070 2 60512 5 54 9 G(n) = ----- F(n) - ----- F(n) - ----- F(n) F(n + 1) + ---- F(n + 1) 6061 6061 6061 6061 54 6 738567 3 2 179291 3 6 - ---- F(n + 1) + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 6061 12122 12122 4817707 4 4835241 8 992665 2 3 + ------- F(n) F(n + 1) - ------- F(n) F(n + 1) - ------ F(n) F(n + 1) 12122 12122 6061 49252 3 3 114559 4 10372467 4 5 + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) - -------- F(n) F(n + 1) 6061 6061 12122 119784 2 4 170814 4 2 + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 6061 6061 11694867 2 7 58152 + -------- 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, 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(6 n - 6 j) F(3 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1168 9 1224 6 5 2 G(n) = ----- F(n) + ---- F(n) F(n + 1) + 232/7 F(n) F(n + 1) 231 77 9220 4 3 604 7 1168 7 244 2 5 - ---- F(n) F(n + 1) - --- F(n + 1) + ---- F(n) + --- F(n) F(n + 1) 231 77 231 231 613 247 5 1780 3 452 4 5 - --- F(n + 1) + --- F(n + 1) + ---- F(n + 1) + --- F(n) F(n + 1) 924 308 231 77 1552 6 11619 6 3 84715 7 2 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 77 77 462 8663 2 3 1629 4 773 8 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 924 28 462 6770 8 - ---- F(n) F(n + 1) 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, 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 ----- \ G(n) = ) F(j) F(6 n - 6 j) F(4 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 6 4 5 G(n) = 1/30 F(n) (816 F(n + 1) + 91875 F(n) F(n + 1) + 3750 F(n) F(n + 1) 4 5 5 + 20955 F(n) F(n + 1) - 593 F(n) + 593 F(n) - 45816 F(n + 1) 9 8 2 3 + 45000 F(n + 1) - 108750 F(n) F(n + 1) - 7830 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, 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(6 n - 6 j) F(4 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1402 3 3855 2 19284 7 19284 10 G(n) = ---- F(n + 1) - ----- F(n + 1) + ----- F(n) - ----- F(n) 6061 12122 6061 6061 111334 5 2 87898 4 3 24449 4 2 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 6061 6061 638 96951 3 7 8529 2 8 119677 9 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) + ------ F(n) F(n + 1) 6061 6061 6061 196813 8 2 1020355 7 3 - ------ F(n) F(n + 1) + ------- F(n) F(n + 1) 12122 6061 817899 6 4 5582 6 3145 5 5 - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 551 551 414 3 4 7736 2 1456 7 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n + 1) 6061 6061 6061 3963 6 + ----- 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, 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 ----- \ 2 G(n) = ) F(j) F(6 n - 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 172 2 4 207 3 431 2 101 2 G(n) = ---- F(n) F(n + 1) - --- F(n) - --- F(n) - --- F(n) F(n + 1) 319 638 638 319 107 2 654 3 3 2851 4 2 - --- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 638 319 638 1241 5 16 5 2 + ---- F(n) F(n + 1) + -- F(n) F(n + 1) + 5/638 F(n + 1) 319 29 3 - 5/638 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, 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(6 n - 5 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 25 2 4 22 3 725 4 2 G(n) = -- F(n) F(n + 1) + -- F(n) F(n + 1) + --- F(n) F(n + 1) 84 21 84 125 5 5 43 2 2 - --- F(n) F(n + 1) - 5/42 F(n) F(n + 1) - -- F(n) F(n + 1) 14 21 6 4 2 2 - 10/7 F(n) - 5/42 + 1/21 F(n) + 5/42 F(n + 1) + 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, 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(6 n - 5 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 453 1424 23485 2 51 5 G(n) = ---- F(n) + ---- F(n + 1) - ----- F(n + 1) - -- F(n + 1) 551 551 1102 19 23595 6 649 2 4455 2 4 141 3 2 + ----- F(n + 1) - --- F(n) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 1102 551 551 19 29480 3 3 3773 97 4 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) + -- F(n) F(n + 1) 551 551 19 26565 5 2 3 - ----- F(n) F(n + 1) + 5/19 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, 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(6 n - 5 j) F(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5666 7 10365 7 9999 2 27445 6 G(n) = ---- F(n + 1) - ----- F(n) + ---- F(n + 1) - ----- F(n) 3781 7562 7562 7562 330 3 3 6512 6 14619 + --- F(n) F(n + 1) - ---- F(n + 1) + ----- F(n) F(n + 1) 19 3781 3781 228525 4 2 87790 4 3 63601 5 2 - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 7562 3781 7562 21290 6 8307 3 121 2 + ----- F(n) F(n + 1) - ---- F(n + 1) - --- F(n) F(n + 1) 3781 7562 38 96135 3 4 35706 5 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 7562 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, 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(6 n - 5 j) F(6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 2 2339 2 2 500 2 4 G(n) = --- F(n) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 11 11 33 2900 3 3 340 5 8 108 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) - 10510/9 F(n + 1) + --- 99 33 11 2 6 2630 3 800 5 - 215/3 F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 33 33 3 5 49741 3 7 - 21730/9 F(n) F(n + 1) - ----- F(n) F(n + 1) + 25715/9 F(n) F(n + 1) 99 114658 4 410 2 6 + ------ F(n + 1) - --- F(n + 1) + 110/9 F(n + 1) 99 33 Proof: Both 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(6 n - 5 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 6821 5 2 1901 6 423 2 2 G(n) = ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 1276 638 638 125 3 5259 4 3 89 2 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 319 638 116 225 6 7089 3 4 71 3 - ---- F(n) F(n + 1) - 5/638 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 1276 1276 319 397 7 3 123 4 + --- F(n) + 5/638 F(n + 1) + --- 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, 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(6 n - 5 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 169 7 111 2 5 215 5 2 G(n) = 1/84 F(n) - --- F(n) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 84 28 14 7 239 3 2 745 3 4 + 5/42 F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 84 84 6 65 2 3 6 - 5/12 F(n) F(n + 1) - -- F(n) F(n + 1) + 60/7 F(n) F(n + 1) 21 41 4 1345 4 3 25 4 - -- F(n) F(n + 1) + ---- F(n) F(n + 1) + -- F(n) F(n + 1) 21 84 84 5 - 5/42 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, 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(6 n - 5 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 79 5 246 2 4 262 4 2 G(n) = -- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 19 19 19 211 2 965 3 55 3 2994 - --- F(n) F(n + 1) + --- F(n) + --- F(n + 1) - ---- F(n) F(n + 1) 19 551 551 551 20395 6 95085 2 5 102 3 3 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - --- F(n) F(n + 1) 551 1102 19 13910 3 4 109395 4 3 688 2 + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) + --- F(n) 551 1102 551 55 2 21303 2 - --- F(n + 1) + ----- F(n) F(n + 1) 551 551 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, 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(6 n - 5 j) F(n + 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 10725 3 349365 5 2 68755 4 4 G(n) = ------ F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 15124 7562 3781 10015 6 161415 6 2 176018 7 - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 398 3781 3781 84840 3 5 11745 3 4 731735 4 3 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 3781 398 15124 42786 2 6 202119 2 5 43721 7 + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) + ----- F(n) 3781 15124 7562 4675 7 136495 4 4475 4 59860 8 + ----- F(n + 1) + ------ F(n) + ---- F(n + 1) - ----- F(n) 15124 7562 7562 3781 725 8 - ---- 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, 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(6 n - 5 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 157 4 102 3 2 G(n) = 5/638 F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 638 319 2444 3 333 4 7353 5 3 + ---- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 638 319 502 7 811 6 2 399 2 6 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 29 638 307 2 3 1167 4 4 201 3 - --- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 319 638 319 5 224 5 414 8 - 5/638 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, 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 ----- \ G(n) = ) F(j) F(6 n - 5 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 61 6 233 2 6 2725 3 G(n) = -- F(n) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 14 42 168 109 3 3 257 4 2 9665 5 3 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 42 21 168 43 7 5 229 3 5 + -- F(n) F(n + 1) + 5/14 F(n) F(n + 1) + --- F(n) F(n + 1) 56 168 1390 6 2 4 7 373 2 + ---- F(n) F(n + 1) + 5/42 F(n + 1) - 5/21 F(n) F(n + 1) - --- F(n) 21 84 487 5 6 37 8 + --- F(n) F(n + 1) - 5/42 F(n + 1) + -- F(n) 42 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, 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(6 n - 5 j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1669 4 111323 3 7 G(n) = ----- F(n) + ------ F(n) F(n + 1) - 200 F(n) F(n + 1) 551 551 5341 2 81613 2 2 2 6 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + 475 F(n) F(n + 1) 551 1102 11063 3 220 3 4 4 4 55 + ----- F(n) F(n + 1) - --- F(n) F(n + 1) - 925/2 F(n) F(n + 1) + --- 551 19 551 15743 2 3 5 1245 4 3 - ----- F(n) F(n + 1) + 50 F(n) F(n + 1) + ---- F(n) F(n + 1) 551 19 515 6 1285 2 5 55 3 1141 3 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) - --- F(n + 1) - ---- F(n) 19 19 551 551 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, 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(6 n - 5 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 18825 4 5 8775 8 117957 4 G(n) = ------ F(n) F(n + 1) - ---- F(n) F(n + 1) + ------ F(n) F(n + 1) 22 22 638 689 2 91 2 549 4 2 127451 + --- F(n) + --- F(n + 1) + --- F(n) F(n + 1) - ------ F(n + 1) 638 638 638 638 63728 5 1225 5 48 6 694 2 4 + ----- F(n + 1) + ---- F(n) - --- F(n + 1) - --- F(n) F(n + 1) 319 638 319 319 1313 5 325 3 6 21329 2 3 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 22 58 21225 2 7 292551 3 2 556 3 3 + ----- 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, 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(6 n - 5 j) F(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 595 6 407 7 3 3 G(n) = ---- F(n) F(n + 1) + --- F(n) - 25 F(n + 1) - 19/4 F(n) 12 84 3341 2 7 1585 3 4 2822 6 3 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 14 21 35 6295 7 2 1235 8 32939 4 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 21 42 140 27869 2 3 499 2 5 1609 5 + ----- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n + 1) 420 14 28 691 1045 7 107 9 12643 8 + --- F(n + 1) + ---- F(n + 1) - --- F(n) - ----- F(n) F(n + 1) 12 42 21 105 131 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, 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(6 n - 5 j) F(3 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 13587 3 660 2 6 472563 4 G(n) = ------ F(n) F(n + 1) - --- F(n) F(n + 1) + ------ F(n) F(n + 1) 551 19 551 5580 8 1829 13720 7 71495 3 - ---- F(n + 1) + ---- + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 19 551 19 551 3 6 11465 3 5 258863 3 2 + 8225/2 F(n) F(n + 1) - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 19 1102 2 7 479423 2 3 32115 2 2 + 525/2 F(n) F(n + 1) - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 1102 551 8 2579 24713 2195789 5 - 5025 F(n) F(n + 1) + ---- F(n) - ----- F(n + 1) - ------- F(n + 1) 551 551 1102 9 159936 4 + 4075/2 F(n + 1) + ------ 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, 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(6 n - 5 j) F(4 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 56910 4 2 1397435 2 8 569130 9 G(n) = ------ F(n) F(n + 1) + ------- F(n) F(n + 1) - ------ F(n) F(n + 1) 319 319 319 2 475605 2 4 15445 2 5 + 15/2 F(n) F(n + 1) - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 638 319 10381 558970 5 6180 6 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 319 26505 4 3 2327245 4 6 113830 3 3 + ----- F(n) F(n + 1) - ------- F(n) F(n + 1) + ------ F(n) F(n + 1) 638 638 319 1805 3 4 50745 3 7 12577 2 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 319 638 2053 2 3 10 1137 3 - ---- F(n) - 5/638 F(n + 1) + 5/638 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, 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(6 n - 5 j) F(4 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4337 29333 5 182386 7 3 G(n) = ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 574 287 287 119339 8 2 1495 3 1495 7 + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 574 42 42 10 3 1465 2 6 + 5/42 F(n + 1) - 5/42 - 163/6 F(n) F(n + 1) - ---- F(n) F(n + 1) 12 63860 2 8 39941 2 4 1369 3 5 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 287 574 14 16724 9 55753 2 1487 8 1217 2 2 + ----- F(n) F(n + 1) - ----- F(n) - ---- F(n) + ---- F(n) F(n + 1) 123 1148 84 28 247 4 4653 10 95017 9 + --- F(n) + ---- F(n) - ----- F(n) F(n + 1) 14 82 861 Proof: Both 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 ----- \ 2 G(n) = ) F(j) F(6 n - 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 G(n) = 1/3 F(n + 1) - 17/2 F(n) F(n + 1) - 2/3 F(n) F(n + 1) 2 2 2 4 3 - 2 F(n) F(n + 1) + 25/2 F(n) F(n + 1) + 4/3 F(n) F(n + 1) 325 6 5 3 3 - --- F(n + 1) - 1/2 + 125/2 F(n) F(n + 1) - 200/3 F(n) F(n + 1) 12 2 2 + 109/4 F(n + 1) + 3/2 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, 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 ----- \ G(n) = ) F(j) F(6 n - 4 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 63 63 2 57 5 581 2 G(n) = ---- F(n + 1) + --- F(n + 1) - --- F(n) - --- F(n) 638 638 638 638 205 2 3 91 2 4 769 3 2 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 638 638 319 966 3 3 843 4 1897 4 2 + --- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 638 319 154 5 695 4 399 5 + --- F(n) F(n + 1) - --- 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, 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(6 n - 4 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 35070 6 579 3 3885 2 7 G(n) = ----- F(n + 1) + ---- F(n) - ---- F(n) - 65 F(n + 1) 551 1102 1102 80955 5 22277 2 86310 3 3 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 551 1102 551 6 11004 3 4 + 150 F(n) F(n + 1) + ----- F(n) F(n + 1) - 160 F(n) F(n + 1) 551 2 5 32445 2 4 1635 2 + 30 F(n) F(n + 1) - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 1102 551 72071 3 70581 2 + ----- 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, 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(6 n - 4 j) F(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 375 2 4 4 4 G(n) = -100 F(n) F(n + 1) + --- F(n) F(n + 1) + 2225/2 F(n) F(n + 1) 22 2 6 3835 3 1125 3 3 - 1150 F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 77 77 3 750 5 4875 4 2 - 975/2 F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 77 154 28075 2 2 765 7 + ----- F(n) F(n + 1) + --- F(n) F(n + 1) + 975/2 F(n) F(n + 1) 154 77 505 4 135 2 15 2 15 + --- F(n) - --- F(n) - -- F(n + 1) + -- 77 77 77 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, 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 ----- \ G(n) = ) F(j) F(6 n - 4 j) F(6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 8647 11004 2 4 74886 4 2 11074 6 G(n) = ---- F(n) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) 3781 3781 3781 3781 836716 3 2 2617506 3 6 + ------ F(n) F(n + 1) - ------- F(n) F(n + 1) 18905 18905 479671 4 5 49969 9 378 9 + ------ F(n) F(n + 1) - ----- F(n) - ---- F(n + 1) 7562 3781 3781 5589867 4 6132 8873127 8 - ------- F(n) F(n + 1) - ---- F(n) F(n + 1) + ------- F(n) F(n + 1) 75620 3781 75620 867279 2 3 859797 2 7 60368 3 3 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 15124 15124 3781 1848 5 378 2 - ---- F(n) F(n + 1) + ---- F(n + 1) 199 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, 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(6 n - 4 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 5 2 G(n) = -7/4 F(n) + 1/6 F(n + 1) + 3/4 F(n) - 161/4 F(n) F(n + 1) 23 3 2 3 4 4 + -- F(n) F(n + 1) - 105/4 F(n) F(n + 1) - 3/2 F(n) F(n + 1) 12 4 2 2 3 + 1/12 F(n) F(n + 1) + 23/2 F(n) F(n + 1) - 17/6 F(n) F(n + 1) 1265 4 3 6 1045 2 5 + ---- F(n) F(n + 1) + 40 F(n) F(n + 1) - ---- F(n) F(n + 1) 12 12 - 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, 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(6 n - 4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 26 2 6 612 3 2999 2 5897 3 G(n) = --- F(n) - 19/2 F(n + 1) + --- F(n) + ---- F(n + 1) - ---- F(n + 1) 319 319 319 58 5 3 3 32465 7 + 41/2 F(n) F(n + 1) - 26 F(n) F(n + 1) + ----- F(n + 1) 319 898 154545 6 259 2 - --- F(n) F(n + 1) - ------ F(n) F(n + 1) - --- F(n) F(n + 1) 319 638 22 2 4 7215 2 5 72720 3 4 + 7/2 F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 638 319 25627 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, 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(6 n - 4 j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 21173 81513 3 13808 2 61565 8 G(n) = - ----- + ----- F(n + 1) - ----- F(n) F(n + 1) + ----- F(n + 1) 1102 2204 551 1102 78529 2 10120 8 6443 7 55310 6 + ----- F(n) F(n + 1) + ----- F(n) + ---- F(n) + ----- F(n) F(n + 1) 2204 551 1102 551 25174 3 67471 2 5 42916 2 2 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 551 1102 551 3042 7 39075 7 159283 6 2 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 29 551 1102 134275 6 39951 4 82395 7 - ------ F(n) F(n + 1) - ----- F(n + 1) - ----- F(n + 1) 2204 1102 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, 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(6 n - 4 j) F(n + 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 318025 9 975 6 54 2 5 G(n) = ------ F(n) - --- F(n) F(n + 1) + -- F(n) F(n + 1) 1386 77 77 151 8 19535 4 5 168881 6 3 - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 1386 231 693 251590 7 2 769075 8 4376 4 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 693 1386 231 2235 6 919 1065 3 + ---- F(n) F(n + 1) - --- F(n + 1) + ---- F(n + 1) 154 99 154 3555 5 2 2410 4 3 1095 7 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n + 1) 154 77 154 23798 5 6568 5 41 7 - ----- F(n) + ---- F(n + 1) + -- F(n) 99 693 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, 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(6 n - 4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 4 7 2 2 G(n) = 500 F(n) F(n + 1) + 650/3 F(n) F(n + 1) + 161/2 F(n) F(n + 1) 2 6 3 3 + 11/2 F(n) F(n + 1) - 1525/3 F(n) F(n + 1) + 22/3 F(n) F(n + 1) 3 5 4 2 3 - 50 F(n) F(n + 1) - 107/6 F(n) F(n + 1) - 217 F(n) F(n + 1) 3 2 4 5 - 22 F(n) F(n + 1) + 49/6 F(n) F(n + 1) - 5 F(n) F(n + 1) 6 2 4 4 - 1/6 F(n + 1) - 5/4 F(n) + 13/4 F(n) + 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, 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(6 n - 4 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 109 3 934 4 63 6074 2 G(n) = 50 F(n) F(n + 1) - --- F(n) - --- F(n) + --- - ---- F(n) F(n + 1) 638 319 638 319 63 3 2737 2 64406 3 - --- F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 638 638 319 6 3 4 7 + 35/2 F(n) F(n + 1) - 35/2 F(n) F(n + 1) - 200 F(n) F(n + 1) 2 5 2 6 6320 3 - 85/2 F(n) F(n + 1) + 475 F(n) F(n + 1) + ---- F(n) F(n + 1) 319 2142 2 2 4 4 4 3 - ---- F(n) F(n + 1) - 925/2 F(n) F(n + 1) + 50 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, 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 ----- \ G(n) = ) F(j) F(6 n - 4 j) F(2 n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 37483 4 1581 98305 5 4 G(n) = ------ F(n) - ---- F(n) - ----- F(n) F(n + 1) 551 1102 551 216419 6 3 92629 7 1224 7 2 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 1102 551 551 16007 8 10473 2 7 8775 3 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 1102 1102 551 27971 3 2 17571 3 5 15643 3 6 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 551 551 1102 3093 9 441 9 441 17571 2 6 + ---- F(n) + ---- F(n + 1) - ---- - ----- F(n) F(n + 1) 1102 1102 1102 1102 21 3 6 2 65079 8 - --- F(n) F(n + 1) - 200 F(n) F(n + 1) + ----- F(n) 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, 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(6 n - 4 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 9 403 2 6 G(n) = 1/6 F(n + 1) + --- F(n) F(n + 1) - 100/3 F(n) F(n + 1) 12 2 5 125 2 4 3 + 215/3 F(n) F(n + 1) - --- F(n) F(n + 1) - 265/3 F(n) F(n + 1) 12 3 3 4 - 1/6 F(n + 1) - 5 F(n) + 2 F(n) + 17/4 F(n) F(n + 1) 3 4 2 3 3 6 + 20 F(n) F(n + 1) + 2531/6 F(n) F(n + 1) + 2975/3 F(n) F(n + 1) 4 4 5 2 7 + 193/4 F(n) F(n + 1) + 905/6 F(n) F(n + 1) - 2555/6 F(n) F(n + 1) 3 2 5 4 - 313/2 F(n) F(n + 1) - 3074/3 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, 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(6 n - 4 j) F(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 17207 8 93197 5 4 57049 6 2 G(n) = ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 6380 1276 3190 268079 6 3 375 7 58461 7 2 - ------ F(n) F(n + 1) - --- F(n) F(n + 1) + ----- F(n) F(n + 1) 1595 58 638 17805 8 2323 2 2 51376 2 3 + ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 1595 1595 4685 2 6 24373 5 3 112 8 63 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + --- F(n) - --- 319 638 319 638 39929 4 25023 3 15644 7 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 3190 3190 1595 17205 63 9 23263 9 - ----- F(n) + --- F(n + 1) + ----- F(n) 1276 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, 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(6 n - 4 j) F(4 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 983 8 8 G(n) = 99/8 F(n) F(n + 1) + 162/5 F(n) F(n + 1) + --- F(n + 1) - 151/9 F(n) 60 689 2 2707 4 6 10 + --- F(n) - ---- F(n + 1) + 4021/9 F(n) + 1/6 F(n + 1) 36 90 10 487 9475 7 3 8 2 - 1373/3 F(n) + --- + ---- F(n) F(n + 1) + 500 F(n) F(n + 1) 36 12 74375 9 9719 5 3 18427 5 5 - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 72 90 72 1871 6 2 755 7 713 5 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 36 18 36 7 475 9 - 340/9 F(n) F(n + 1) + --- F(n) F(n + 1) 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, 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(6 n - 4 j) F(4 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 11867 135 9 305925 6 687475 8 2 G(n) = ----- F(n + 1) - --- F(n) - ------ F(n + 1) - ------ F(n) F(n + 1) 319 319 319 319 148759 4 60425 9 339841 2 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ------ F(n) 1276 29 638 344675 10 29559 2 3 611913 2 - ------ F(n) + ----- F(n) F(n + 1) + ------ F(n + 1) 638 638 638 816525 5 56365 8 203925 9 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 319 1276 319 328175 2 4 16625 8 48363 2 7 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 638 638 638 39657 3 2 759675 2 8 200023 7 2 - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 1276 638 1276 204350 7 3 23797 5 + ------ 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, 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 ----- \ 2 G(n) = ) F(j) F(6 n - 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 80 5 4 135 4 2 G(n) = -- F(n) F(n + 1) + F(n) F(n + 1) - --- F(n) F(n + 1) 11 11 4 118 5 19 6 - 8/11 F(n) F(n + 1) - 4 F(n) F(n + 1) + --- F(n) F(n + 1) - -- F(n) 11 11 5 5 45 6 18 71 - 3/5 F(n) + 6/5 F(n + 1) - -- F(n + 1) + -- F(n) - -- F(n + 1) 11 55 55 46 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, 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(6 n - 3 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 60 3 128 3 1216 2 128 2 G(n) = ---- F(n) + --- F(n + 1) - ---- F(n) - --- F(n + 1) 319 319 319 319 7839 4 3 6720 5 6448 5 2 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 16 68 6 1088 2 + -- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 29 319 319 1731 2 5 3440 3 3 3122 3 4 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 8520 4 2 1560 2 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, 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(6 n - 3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1125 5 6609 3 8 G(n) = ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - 2225/2 F(n + 1) 14 14 2 6 981 2 27 2 975 6 - 75/2 F(n) F(n + 1) - --- F(n + 1) - -- F(n) + --- F(n + 1) 28 14 28 3 3 3 111 15467 4 - 467/7 F(n) F(n + 1) + 600/7 F(n) F(n + 1) + --- + ----- F(n + 1) 14 14 7 2 2 225 2 4 + 5425/2 F(n) F(n + 1) + 381/2 F(n) F(n + 1) - --- F(n) F(n + 1) 14 3 5 153 - 2325 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, 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(6 n - 3 j) F(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 425 6 195 2 4 1055 5 1555 6 G(n) = ---- F(n + 1) + --- F(n) F(n + 1) + ---- F(n + 1) + ---- F(n) 551 29 1102 551 7065 5 1165 8370 4 2 3435 2 7 - ---- F(n) - ---- F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 551 1102 551 551 158395 3 6 87055 4 480 2 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + --- F(n + 1) 551 1102 551 340940 4 5 9835 3 2 8460 3 3 - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 551 38 551 137715 5 4 723205 6 3 5790 5 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 1102 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, 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 ----- \ G(n) = ) F(j) F(6 n - 3 j) F(6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 8 G(n) = 1/3 F(n) (2446 F(n) F(n + 1) - 55575 F(n) F(n + 1) 2 7 2 3 9 + 2550 F(n) F(n + 1) + 48 F(n) - 4686 F(n) F(n + 1) + 22575 F(n + 1) 3 6 4 + 45750 F(n) F(n + 1) - 460 F(n + 1) + 9467 F(n) F(n + 1) 5 - 22115 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, 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(6 n - 3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 32 370 6 107 2 G(n) = --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 11 11 11 855 2 5 3 3 595 3 4 + --- F(n) F(n + 1) - 7 F(n) F(n + 1) + --- F(n) F(n + 1) 11 22 1035 4 3 5 31 3 2 - ---- F(n) F(n + 1) + 3/2 F(n) F(n + 1) + -- F(n) - 1/11 F(n + 1) 11 22 13 2 3 783 2 2 4 + -- F(n) + 1/11 F(n + 1) + --- F(n) F(n + 1) - 6 F(n) F(n + 1) 22 22 4 2 + 19/2 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, 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 ----- \ G(n) = ) F(j) F(6 n - 3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 9633 6 7954 6 1905 6 2 G(n) = ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 29 2240 7 1568 7 19986 2 5 + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 29 29 319 7048 3 6446 4 490 3 606 8 - ---- F(n) F(n + 1) + ---- F(n + 1) + --- F(n + 1) + --- F(n) 145 145 319 29 23747 5 3 3406 8 6560 1808 7 - ----- F(n) F(n + 1) - ---- F(n + 1) - ---- + ---- F(n) 145 145 319 319 618 7 22904 5 2 8165 2 - --- 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, 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(6 n - 3 j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 557 5 3781 3 6 359 9 79 7 G(n) = ---- F(n + 1) - ---- F(n) F(n + 1) - --- F(n) + -- F(n) 112 112 28 28 551 9 387 6 18647 7 2 + --- F(n + 1) + --- F(n) F(n + 1) - ----- F(n) F(n + 1) 224 28 56 1973 3 2 439 4 3 20901 4 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) + ----- F(n) F(n + 1) 112 14 224 6 75699 6 3 881 8 - 87/7 F(n) F(n + 1) + ----- F(n) F(n + 1) - --- F(n) F(n + 1) 224 28 39 3 4 5 2 183 3 + -- F(n) F(n + 1) + 162/7 F(n) F(n + 1) + --- F(n + 1) 28 28 7 611 - 27/4 F(n + 1) + --- F(n + 1) 224 Proof: Both 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(6 n - 3 j) F(n + 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2562 2 5 11879 7 22865 6 28991 3 G(n) = ---- F(n) F(n + 1) + ----- F(n) + ----- F(n) - ----- F(n) 551 551 1102 1102 40230 6 5925 4 3 121359 4 6 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 551 551 551 2891 9 86556 4 2 76554 5 + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 1102 551 551 68605 5 2 524091 5 5 351903 7 3 - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 1102 1102 551 377969 8 2 2041 2 6876 2 4 + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 1102 1102 551 14601 3 7 55 3 55 10 + ----- F(n) F(n + 1) + --- F(n + 1) - --- F(n + 1) 1102 551 551 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, 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(6 n - 3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 104 7 7 69 8 227 3 8 G(n) = --- F(n) - 1/11 F(n + 1) - -- F(n) - --- F(n) + 1/11 F(n + 1) 11 22 22 100 4 3 621 5 3 260 6 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 11 11 11 51 2 5 218 6 2 219 2 6 - -- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 22 11 22 2 2 15 2 929 4 4 - 10 F(n) F(n + 1) - -- F(n) F(n + 1) - --- F(n) F(n + 1) 11 22 389 5 2 21 3 160 7 - --- F(n) F(n + 1) + -- F(n) 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, 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(6 n - 3 j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3024 8 9 26914 3 G(n) = - ---- + 1500 F(n + 1) - 1500 F(n + 1) + ----- F(n) F(n + 1) 319 319 18236 4 2 6 43204 5 - ----- F(n) F(n + 1) + 25 F(n) F(n + 1) + ----- F(n + 1) 29 29 201590 3 7 2 7 + ------ F(n) F(n + 1) - 3650 F(n) F(n + 1) - 25 F(n) F(n + 1) 319 2493 3 2 7228 2 3 7257 2 2 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 29 29 29 8 3 5 3 6 + 3650 F(n) F(n + 1) + 3150 F(n) F(n + 1) - 3150 F(n) F(n + 1) 475604 4 166 3384 - ------ 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, 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(6 n - 3 j) F(2 n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 10 2 3 3 5 G(n) = 3/14 F(n + 1) + 83/4 F(n) + 325 F(n) F(n + 1) + 75 F(n) F(n + 1) 7 9 - 3/14 - 325 F(n) F(n + 1) + 71222/7 F(n) F(n + 1) 3 3 1689 2 2 59543 2 4 - 2078 F(n) F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 14 14 2 6 2 8 3 + 1525/2 F(n) F(n + 1) - 175089/7 F(n) F(n + 1) + 232/7 F(n) F(n + 1) 14531 3 7 4 2 4 4 + ----- F(n) F(n + 1) + 7503/7 F(n) F(n + 1) - 750 F(n) F(n + 1) 14 4 6 139655 5 + 144989/7 F(n) F(n + 1) - 1402/7 F(n) F(n + 1) - ------ F(n) F(n + 1) 14 127 4 - --- F(n) 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, 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(6 n - 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 1341 6 2 35 2 2 17 8 G(n) = ---- F(n) F(n + 1) - -- F(n) F(n + 1) + -- F(n) - 1/11 22 11 11 41 3 2487 5 2 3 - -- F(n) F(n + 1) + ---- F(n) - 9/2 F(n) F(n + 1) 22 22 232 3 375 3 2 45 8 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + -- F(n) F(n + 1) 11 22 22 258 4 4 222 4 5 2293 6 3 - --- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 11 22 343 7 3549 7 2 5553 8 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 22 22 22 893 5 3 9 9 - --- F(n) F(n + 1) - 217/2 F(n) + 1/11 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, 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(6 n - 3 j) F(3 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 128 10 272 10 277714 5 233503 4 2 G(n) = --- F(n + 1) + --- F(n) + ------ F(n) + ------ F(n) F(n + 1) 319 319 1595 1595 22072 4 5 159958 6 4 151464 7 2 - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 145 319 319 153193 7 3 123467 8 104563 8 2 - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 319 319 1595 4546 9 3495 4 3150 5 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 319 319 319 2658 9 79064 3 7 73153 4 + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 1595 1595 2895 8 36191 3 3 25024 3 6 - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 1595 319 128 9 253554 9 - --- F(n + 1) - ------ F(n) 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, 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(6 n - 3 j) F(4 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3803 5 1779 9 169 10 G(n) = ----- F(n) + ---- F(n) + 9/22 F(n) F(n + 1) - --- F(n) 110 55 22 9 823 4 3 7 - 1/11 F(n + 1) + --- F(n) F(n + 1) + 75/2 F(n) F(n + 1) 55 8807 9 7925 8 2 720 8 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 110 22 11 4306 7 3 2119 7 2 6956 5 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 22 55 39 5 213 3 6 17 8 + -- F(n) F(n + 1) - --- F(n) F(n + 1) - -- F(n) F(n + 1) 22 11 11 3593 4 6 2119 4 5 2689 4 2 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 55 110 110 10 + 1/11 F(n + 1) - 1/11 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, 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 ----- \ G(n) = ) F(j) F(6 n - 2 j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 192 6 96 7 4 3 G(n) = ---- F(n) F(n + 1) + -- F(n + 1) + 30 F(n) F(n + 1) 11 11 585 4 2 138 5 2 41 6 - --- F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 11 159 5 78 2 69 6 + 6/11 F(n) F(n + 1) + --- F(n) F(n + 1) + -- F(n + 1) - -- F(n) 11 11 22 6 183 3 3 2 - 15/2 F(n + 1) - --- F(n + 1) + 3/22 F(n) - 1/22 F(n) F(n + 1) 22 120 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, 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(6 n - 2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 232 3 5 179 359 8 G(n) = ---- F(n) F(n + 1) + 16/3 F(n) F(n + 1) - --- + --- F(n) 15 24 24 13 8 451 4 5 + --- F(n + 1) + --- F(n + 1) + 1/6 F(n) F(n + 1) + 43/6 F(n) F(n + 1) 120 60 4 2 1207 4 4 1297 5 3 - 25/2 F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 24 10 6 2 7 6 43 6 + 351/4 F(n) F(n + 1) - 13/2 F(n) F(n + 1) - 3/2 F(n) - -- F(n + 1) 12 41 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, 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(6 n - 2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1971 2 17874 2 2 7 G(n) = ---- F(n) + ----- F(n + 1) - 625 F(n) F(n + 1) 638 319 3 6 4 8 - 11325/2 F(n) F(n + 1) - 2437/2 F(n) F(n + 1) + 14525/2 F(n) F(n + 1) 2 3 15255 2 4 513 + 736 F(n) F(n + 1) + ----- F(n) F(n + 1) - --- F(n) F(n + 1) 638 29 5 9 40770 5 + 2810 F(n + 1) - 5825/2 F(n + 1) + ----- F(n) F(n + 1) 319 3 2 44190 3 3 32666 7713 - 479 F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n + 1) - ---- F(n) 319 319 638 35685 6 - ----- 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, 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(6 n - 2 j) F(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 8 3 3 G(n) = 4885/6 F(n + 1) - 3275 F(n) F(n + 1) - 9580/3 F(n) F(n + 1) 3 7 9 - 29650 F(n) F(n + 1) - 455/3 F(n) F(n + 1) + 38025 F(n) F(n + 1) 2 4 6 2 10 + 3975 F(n) F(n + 1) + 86615/6 F(n + 1) + 15 F(n) - 15250 F(n + 1) 5 - 5745 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, 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(6 n - 2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2821 2 3 G(n) = 651/2 F(n + 1) - 1/11 + ---- F(n) F(n + 1) - 6 F(n) F(n + 1) 22 67 3 3627 3 17305 6 2 6 + -- F(n) - ---- F(n + 1) - ----- F(n) F(n + 1) + 50 F(n) F(n + 1) 11 11 22 7 835 2 2 2 + 775 F(n) F(n + 1) - --- F(n) F(n + 1) + 73/2 F(n) F(n + 1) 22 1035 2 5 8055 3 4 3 5 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - 725 F(n) F(n + 1) 22 11 3 7245 7 8 - 251/2 F(n) F(n + 1) + ---- F(n + 1) - 325 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, 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(6 n - 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 3 2 431 3 4 G(n) = -1/6 F(n + 1) - 1079/3 F(n) F(n + 1) + --- F(n) F(n + 1) 12 4 4 5 4 + 344/3 F(n) F(n + 1) + 29251/6 F(n) F(n + 1) - 13655/6 F(n) F(n + 1) 8 2 2 3 + 13723/6 F(n) F(n + 1) - 115/6 F(n) F(n + 1) + 2845/3 F(n) F(n + 1) 2 7 145 37 3 6 - 16613/3 F(n) F(n + 1) - --- F(n) + -- F(n) - 383/6 F(n) F(n + 1) 12 12 2 5 1987 4 3 3 6 + 569/4 F(n) F(n + 1) - ---- F(n) F(n + 1) - 223/6 F(n) F(n + 1) 12 773 2 9 + --- F(n) F(n + 1) + 1/6 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, 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(6 n - 2 j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 64331 2 2923 3 166345 4 3 G(n) = ------ F(n) F(n + 1) - ---- F(n) + ------ F(n) F(n + 1) 638 638 638 70830 2 5 1662053 2 4 643 2 - ----- F(n) F(n + 1) + ------- F(n) F(n + 1) + --- F(n) F(n + 1) 319 638 22 1961913 9 31520 6 1904367 5 + ------- F(n) F(n + 1) + ----- F(n) F(n + 1) - ------- F(n) F(n + 1) 319 319 319 56723 4876681 2 8 7703787 4 6 - ----- F(n) F(n + 1) - ------- F(n) F(n + 1) + ------- F(n) F(n + 1) 319 319 638 467398 3 3 40245 3 4 279891 4 2 - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 319 638 319 12493 2 63 3 374762 3 7 63 10 + ----- F(n) + --- F(n + 1) + ------ F(n) F(n + 1) - --- F(n + 1) 638 638 319 638 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, 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(6 n - 2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 516 3 5 G(n) = 9/22 F(n + 1) - 3/11 F(n) F(n + 1) + --- F(n) F(n + 1) 11 2183 4 4 166 2 7 345 2 6 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 22 11 22 523 3 6 540 5 97 8 1085 9 - --- F(n) F(n + 1) - --- F(n) - 9/22 - -- F(n) + ---- F(n) 11 11 11 22 8 1120 4 5 1328 5 3 + 5/22 F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 11 1585 5 4 2217 6 2 527 7 - ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 22 22 11 1063 7 2 1189 8 - ---- 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, 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 ----- \ G(n) = ) F(j) F(6 n - 2 j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 10 6 595 4 4997 10 G(n) = - 1/6 + 1/6 F(n + 1) + 436 F(n) - --- F(n) - ---- F(n) 12 12 8 7 6 2 + 271/6 F(n) + 123 F(n) F(n + 1) - 100 F(n) F(n + 1) 4 2 3 7 - 5/6 F(n) F(n + 1) + 566/3 F(n) F(n + 1) - 167/3 F(n) F(n + 1) 3 5 2 8 2 6 + 23 F(n) F(n + 1) + 39/4 F(n) F(n + 1) - 23/3 F(n) F(n + 1) 6 4 5 5 5 3 + 1049/4 F(n) F(n + 1) + 13/3 F(n) F(n + 1) - 115/3 F(n) F(n + 1) 9 8 2 - 1981/2 F(n) F(n + 1) + 3401/6 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, 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 ----- \ G(n) = ) F(j) F(6 n - 2 j) F(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 454137 5 91327 6 182331 8 G(n) = ------ F(n + 1) - ----- F(n + 1) + ------ F(n) F(n + 1) 4510 902 902 1329213 7 2 3106919 6 3 25939 5 + ------- F(n) F(n + 1) - ------- F(n) F(n + 1) - ----- F(n) F(n + 1) 4510 4510 451 10 130476 4 28953 9 - 7/22 F(n) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 2255 451 94607 8 2 235002 8 132372 7 3 - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 902 2255 451 46768 10 466401 9 43187 + ----- F(n + 1) - ------ F(n + 1) + ----- F(n) F(n + 1) 451 4510 902 91420 9 920 2 622277 6 4 - ----- F(n) F(n + 1) - --- F(n + 1) + ------ F(n) F(n + 1) 451 451 902 215277 10419 142181 9 - ------ F(n) + ----- F(n + 1) + ------ F(n) 4510 4510 2255 Proof: Both 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(6 n - j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 5 2 G(n) = -39/2 F(n + 1) + 15/2 F(n) - 270 F(n) F(n + 1) - 15 F(n) F(n + 1) 3 3 7 6 + 290 F(n) F(n + 1) + 20 F(n + 1) + 37 F(n) F(n + 1) + 235/2 F(n + 1) 2 4 2 5 2 - 105/2 F(n) F(n + 1) + 57 F(n) F(n + 1) + 47/2 F(n) F(n + 1) 6 7 2 2 - 65 F(n) F(n + 1) - 6 F(n) - 13/2 F(n) - 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, 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 ----- \ G(n) = ) F(j) F(6 n - j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 2 4 3 G(n) = 8 F(n + 1) - 3 F(n) + 75/2 F(n) F(n + 1) - 5143/6 F(n) F(n + 1) 6 4 4 2 6 - 25/3 F(n + 1) + 385/2 F(n) + 3892/3 F(n + 1) - 75/2 F(n) F(n + 1) 4 2 3 7 - 125/2 F(n) F(n + 1) + 952/3 F(n) F(n + 1) + 5425/2 F(n) F(n + 1) 3 3 3 5 + 25/3 F(n) F(n + 1) - 2325 F(n) F(n + 1) + 17 F(n) F(n + 1) 8 - 2225/2 F(n + 1) - 369/2 Proof: Both 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 ----- \ G(n) = ) F(j) F(6 n - j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 9 905 6 1120 3 3 G(n) = -5825/2 F(n + 1) - --- F(n + 1) - ---- F(n) F(n + 1) 22 11 1035 5 135 1121 907 2 + ---- F(n) F(n + 1) - --- F(n) + ---- F(n + 1) + --- F(n + 1) 11 11 11 22 2 7 3 2 25 2 - 625 F(n) F(n + 1) - 957/2 F(n) F(n + 1) + -- F(n) 11 8 2 3 195 2 4 + 14525/2 F(n) F(n + 1) + 736 F(n) F(n + 1) + --- F(n) F(n + 1) 11 3 6 4 - 11325/2 F(n) F(n + 1) - 13 F(n) F(n + 1) - 2439/2 F(n) F(n + 1) 5 + 5621/2 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, 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 ----- \ G(n) = ) F(j) F(6 n - j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 6 10 2 8 G(n) = 15 F(n) - 761/2 F(n + 1) - 425 F(n + 1) - 3275 F(n) F(n + 1) 9 5 - 149 F(n) F(n + 1) + 2445 F(n) F(n + 1) + 165 F(n) F(n + 1) 6 4 3 3 2 4 + 2965 F(n) F(n + 1) - 3172 F(n) F(n + 1) + 1006 F(n) F(n + 1) 2 + 1611/2 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, 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 ----- \ G(n) = ) F(j) F(6 n - j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 2 2 2 G(n) = -85/2 F(n + 1) - 59/2 F(n) F(n + 1) + 81/2 F(n) F(n + 1) 2 5 4 3 3 + 545/2 F(n) F(n + 1) - 340 F(n) F(n + 1) - 249/2 F(n) F(n + 1) 8 2 2 6 - 325 F(n + 1) + 1 + 219/2 F(n) F(n + 1) + 50 F(n) F(n + 1) 3 5 6 3 - 725 F(n) F(n + 1) - 45/2 F(n) F(n + 1) - 10 F(n) F(n + 1) 4 3 3 7 + 649/2 F(n + 1) + 4 F(n) + 42 F(n + 1) + 775 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, 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(6 n - j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 2 3 9 G(n) = 13/6 F(n) + 89/6 F(n) F(n + 1) - 1/3 F(n + 1) - 529/6 F(n) 2 2377 4 823 8 - 37/6 F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 15 10 2 3 2 7 3 4 + 892/5 F(n) F(n + 1) - 539/3 F(n) F(n + 1) + 85/2 F(n) F(n + 1) 7 2 8 - 931/3 F(n) F(n + 1) - 1435/6 F(n) F(n + 1) + 76 F(n) + 1/3 F(n + 1) 6 2 5 9614 6 3 + 20/3 F(n) F(n + 1) - 36 F(n) F(n + 1) + ---- F(n) F(n + 1) 15 6 - 155/6 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, 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(6 n - j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 10 221 10 56 7 G(n) = 1/11 F(n + 1) - 1/11 F(n + 1) + --- F(n) - -- F(n) 11 11 364 3 7 27 9 4531 8 2 + --- F(n) F(n + 1) + -- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 11 22 241 2 8 3 4 232 6 - --- F(n) F(n + 1) - 45/2 F(n) F(n + 1) + --- F(n) F(n + 1) 22 11 993 4 6 1073 9 805 4 3 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 22 11 22 2496 7 3 159 2 5 831 5 2 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 11 22 22 2643 6 4 19 6 5 5 + ---- F(n) F(n + 1) - -- 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, 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(6 n - j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 9 2 7 4 G(n) = 1/2 F(n + 1) - 1/2 - 445/2 F(n) F(n + 1) - 10 F(n) 8 5 2 2 - 5/2 F(n) F(n + 1) + 1/2 F(n) - 487/2 F(n) F(n + 1) 2 3 3 3 2 + 485/2 F(n) F(n + 1) + 69 F(n) F(n + 1) - 133/2 F(n) F(n + 1) 3 5 3 6 4 - 501 F(n) F(n + 1) + 1005/2 F(n) F(n + 1) + 17/2 F(n) F(n + 1) 4 5 7 2 6 + 395/2 F(n) F(n + 1) + F(n) F(n + 1) + 223 F(n) F(n + 1) 4 4 5 4 5 3 - 198 F(n) F(n + 1) - 1303/2 F(n) F(n + 1) + 651 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, 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(6 n - j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 3 43256 2 8 G(n) = 50 F(n) F(n + 1) - 50 F(n) F(n + 1) - 1/3 - ----- F(n) - 75/2 F(n) 45 4 301421 9 2 2 + 98/3 F(n) + ------ F(n) F(n + 1) + 197/3 F(n) F(n + 1) 135 2 4 2 6 16408 5 - 14369/9 F(n) F(n + 1) - 175 F(n) F(n + 1) - ----- F(n) F(n + 1) 15 153974 1836409 8 2 1314457 9 - ------ F(n) F(n + 1) + ------- F(n) F(n + 1) + ------- F(n) F(n + 1) 135 270 270 563957 2 8 3 797786 3 3 - ------ F(n) F(n + 1) - 122/3 F(n) F(n + 1) - ------ F(n) F(n + 1) 135 135 3 5 10 88327 10 + 150 F(n) F(n + 1) + 1/3 F(n + 1) + ----- F(n) 90 Proof: Both 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(6 n - j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 2 7 2 8 G(n) = 2529/2 F(n) F(n + 1) + 825 F(n) F(n + 1) - 14851/2 F(n) F(n + 1) 3 2 521 2 3 8 + 939/2 F(n) F(n + 1) + --- F(n) F(n + 1) - 2169/5 F(n) F(n + 1) 10 9 4 2 4 6 + 3074 F(n) F(n + 1) + 291/2 F(n) F(n + 1) + 13201/2 F(n) F(n + 1) 13201 6 3 5 - ----- F(n) F(n + 1) - 31/2 F(n) F(n + 1) - 3061 F(n) F(n + 1) 10 4 3 7 3 3 - 287/2 F(n) F(n + 1) - 99 F(n) F(n + 1) - 469 F(n) F(n + 1) 2 9 10 + 29/2 F(n) + 1/2 F(n) - 1/2 F(n + 1) + 1/2 F(n + 1) 3 6 4223 4 + 99 F(n) F(n + 1) + ---- 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, 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 ----- \ 3 G(n) = ) F(j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 3 G(n) = 1/2 - 1/2 F(n + 1) + 3/2 F(n) F(n + 1) - 3/2 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, 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 ----- \ 2 G(n) = ) F(j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = F(n) (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, 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 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 G(n) = - 1/11 + 1/11 F(n + 1) + 6/11 F(n) + 4/11 F(n) F(n + 1) 12 3 2 29 4 27 5 - -- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) 11 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, 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 ----- \ 2 G(n) = ) F(j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 2 3 3 G(n) = 19/6 F(n) - 1/6 + 1/4 F(n + 1) - 9/2 F(n) F(n + 1) 5 2 4 4 2 - 15/2 F(n) F(n + 1) + 5/3 F(n) F(n + 1) + 43/6 F(n) F(n + 1) 6 - 1/12 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, 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 ----- \ 2 G(n) = ) F(j) F(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 55500 7 8100 2 125 2 5 G(n) = ------ F(n + 1) - 5/638 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 319 111005 3 3185 3 117500 3 4 + ------ F(n + 1) - ---- F(n) - ------ F(n) F(n + 1) 638 638 319 134775 6 47315 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, 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 ----- \ 2 G(n) = ) F(j) F(6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 4 3 G(n) = -790 F(n + 1) + 782 F(n + 1) + 8 - 62 F(n) F(n + 1) 2 6 2 2 3 5 - 235/3 F(n) F(n + 1) + 463/3 F(n) F(n + 1) - 1610 F(n) F(n + 1) 3 7 - 1037/3 F(n) F(n + 1) + 5825/3 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, 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 ----- \ 2 G(n) = ) F(j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 2 G(n) = - 1/2 + 1/2 F(n + 1) - 1/2 F(n) + F(n) F(n + 1) - 3/2 F(n) F(n + 1) 3 4 - 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, 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 ----- \ 2 G(n) = ) F(j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 36 2 5 2 G(n) = -- F(n) - 3/11 F(n) - 1/11 F(n + 1) + 1/11 F(n + 1) 11 4 5 2 3 - 3/11 F(n) F(n + 1) + 9/11 F(n) F(n + 1) + 6/11 F(n) F(n + 1) 54 2 4 3 2 60 3 3 - -- F(n) F(n + 1) + 3/11 F(n) F(n + 1) + -- F(n) F(n + 1) 11 11 4 106 4 2 5 - 6/11 F(n) F(n + 1) + --- F(n) F(n + 1) - 14 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, 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 ----- \ 2 G(n) = ) F(j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 4 6 G(n) = -1/12 F(n) F(n + 1) + 1/6 F(n) F(n + 1) + 77/3 F(n) F(n + 1) 2 4 2 3 - 1/4 F(n) F(n + 1) + 1/12 F(n) F(n + 1) - 1/6 F(n) F(n + 1) 61 3 3 23 2 5 - 1/6 F(n + 1) - -- F(n) + 1/6 F(n + 1) + -- F(n) F(n + 1) 12 12 3 4 5 47 4 3 - 13/6 F(n) F(n + 1) + 1/12 F(n) + -- F(n) F(n + 1) 12 289 5 2 - --- 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, 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 ----- \ 2 G(n) = ) F(j) F(n + 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 22499 8 4375 7 34055 6 2 2573 G(n) = ------ F(n + 1) - ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - ---- 3190 319 638 638 3867 8 57 92819 5 3 + ---- F(n) - 5/638 F(n + 1) - --- F(n) - ----- F(n) F(n + 1) 319 638 1595 1306 3 4946 7 35389 4 - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n + 1) 145 319 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, 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(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 G(n) = -1/2 F(n + 1) + 1/2 F(n + 1) + 3/2 F(n) - F(n) F(n + 1) + F(n) 2 3 3 2 4 5 - 1/2 F(n) F(n + 1) + 5/2 F(n) F(n + 1) + F(n) F(n + 1) - 9/2 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, 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 ----- \ 2 G(n) = ) F(j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 835 6 1965 2 5 1905 4 3 G(n) = --- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 11 11 11 180 3 4 823 2 2 2 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - 1/11 F(n + 1) + 2/11 F(n) 11 11 57 3 3 289 2 - -- F(n) + 1/11 F(n + 1) - 6/11 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, 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 ----- \ 2 G(n) = ) F(j) F(2 n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1781 8 3 3 8045 4 4 G(n) = ----- F(n + 1) + 1/2 F(n) F(n + 1) - ---- F(n) F(n + 1) 12 12 5 6 6 2 4 - 1/3 F(n) F(n + 1) + 1/12 F(n + 1) - 1/4 F(n) - 2/3 F(n) F(n + 1) 4 2 2 6 3 5 + 2/3 F(n) F(n + 1) + 1790/3 F(n) F(n + 1) + 995/3 F(n) F(n + 1) 5 3 4 4 2 - 1993/3 F(n) F(n + 1) + 313/2 F(n) + 1781/6 F(n + 1) - 1/4 F(n + 1) 3 + 250 F(n) F(n + 1) - 593/4 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, 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 ----- \ 2 G(n) = ) F(j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 6 5 3 3 2 G(n) = -91/2 F(n + 1) + 112 F(n) F(n + 1) - 94 F(n) F(n + 1) + 45 F(n + 1) 2 3 3 + 9/2 F(n) - 3/2 F(n) + 1/2 F(n + 1) - 19 F(n) F(n + 1) 2 4 2 2 - 7/2 F(n) F(n + 1) - 3/2 F(n) F(n + 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, 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 ----- \ 2 G(n) = ) F(j) F(3 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 49 8 3 10 4 3 25 6 G(n) = --- F(n + 1) + 7/22 F(n + 1) - -- F(n) F(n + 1) - -- F(n) F(n + 1) 132 11 22 17 2 5 2 20 2 5 + -- F(n) F(n + 1) - 9/11 F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 19 4 4 809 5 3 1607 6 2 + -- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 44 33 33 964 7 1157 8 227 2 2 4 - --- F(n) F(n + 1) + ---- F(n) - --- F(n) F(n + 1) + 1/33 F(n + 1) 33 132 66 7 7 41 - 5/11 F(n) - 9/22 F(n + 1) - --- 132 Proof: Both 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 ----- \ 2 G(n) = ) F(j) F(4 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 3 4 3 G(n) = -15/2 F(n) + 1/2 - 1/2 F(n + 1) + 2 F(n) - F(n) F(n + 1) 2 6 2 2 + 21/4 F(n) F(n + 1) - 11/4 F(n) F(n + 1) + 13/2 F(n) F(n + 1) 3 4 3 6 - 7 F(n) F(n + 1) - 35/2 F(n) F(n + 1) + 39/2 F(n) F(n + 1) 3 4 5 2 + 95/4 F(n) F(n + 1) - 85/4 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, 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 ----- \ 2 G(n) = ) F(j) F(5 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 4 8 5 3 G(n) = -183 + 195 F(n) + 364 F(n + 1) - 363/2 F(n + 1) - 778 F(n) F(n + 1) 5 5 3 5 + 1/2 F(n) + 3/2 F(n + 1) - 9/2 F(n) - F(n + 1) + 470 F(n) F(n + 1) 4 2 6 3 + 45/2 F(n) F(n + 1) + 715 F(n) F(n + 1) + 302 F(n) F(n + 1) 3 2 4 4 - 35/2 F(n) F(n + 1) - 905 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 ----- \ 2 G(n) = ) F(j) F(2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 19 2 3 G(n) = 9/22 - 9/22 F(n + 1) + 1/22 F(n) + -- F(n) F(n + 1) 22 57 3 2 69 4 16 5 - -- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) 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, 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 ----- \ G(n) = ) F(j) F(2 j) F(3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 3 4 2 G(n) = -1/6 F(n + 1) + 1/6 - 17/3 F(n) F(n + 1) + 41/4 F(n) F(n + 1) 11 2 4 6 5 + -- F(n) F(n + 1) + 11/6 F(n) - 15/2 F(n) F(n + 1) 12 5 + 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, 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(2 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 63 3 8530 3 4 6360 2 G(n) = --- F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 638 319 319 46925 2 5 50135 4 3 20440 6 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 319 63 41109 2 1851 3 - --- - ----- F(n) 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, 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(2 j) F(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 3 3 5 G(n) = 325/3 F(n) F(n + 1) + 15 F(n) F(n + 1) + 1675/6 F(n) F(n + 1) 5 3 4 4 - 2275/6 F(n) F(n + 1) + 85/2 F(n + 1) + 45/2 F(n) 4 4 8 - 275/6 F(n) F(n + 1) - 35/2 - 25 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, 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(2 j) F(6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3816721 2 3 2962721 3 2 G(n) = ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 6061 12122 471761 4 54 9 54 17914925 4 + ------ F(n) F(n + 1) + ---- F(n + 1) - ---- - -------- F(n) F(n + 1) 6061 6061 6061 12122 48434 395391 3 6 37823883 4 5 - ----- F(n) + ------ F(n) F(n + 1) + -------- F(n) F(n + 1) 6061 12122 12122 18009259 8 43830983 2 7 + -------- F(n) F(n + 1) - -------- F(n) 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, 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(2 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 5 21 6 2 G(n) = -9/22 F(n + 1) - 3/11 F(n) + -- F(n) + 4/11 F(n) + 9/22 F(n + 1) 11 37 3 3 4 181 4 2 - -- F(n) F(n + 1) - 6/11 F(n) F(n + 1) + --- F(n) F(n + 1) 11 22 79 5 4 5 - -- F(n) F(n + 1) - 3/11 F(n) F(n + 1) + 7/22 F(n) F(n + 1) 11 2 3 3 2 + 6/11 F(n) F(n + 1) + 3/11 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, 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 ----- \ G(n) = ) F(j) F(2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 2 3 4 G(n) = -1/12 F(n) F(n + 1) + 1/6 F(n) F(n + 1) - 1/6 F(n) F(n + 1) 3 5 35 3 2005 4 3 - 1/6 F(n + 1) - 1/12 F(n) - -- F(n) + ---- F(n) F(n + 1) 12 12 3 4 3 2 1835 2 5 - 125/4 F(n) F(n + 1) + 1/12 F(n) F(n + 1) - ---- F(n) F(n + 1) 12 2 805 6 2 + 119/6 F(n) F(n + 1) + --- F(n) F(n + 1) - 401/6 F(n) F(n + 1) 12 + 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, 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(2 j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 13725 3 5 101714 3 226175 4 4 G(n) = ------ F(n) F(n + 1) - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 319 319 319 11265 3 242225 2 6 79231 2 2 - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 319 319 638 101575 7 175 63 1476 4 63 + ------ F(n) F(n + 1) + --- F(n) - --- F(n + 1) + ---- F(n) + --- 319 638 638 319 638 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, 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(2 j) F(n + 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 4 2 3 G(n) = 18763/6 F(n + 1) - 8153/6 F(n) F(n + 1) + 720 F(n) F(n + 1) 8 2 7 3 2 + 15865/2 F(n) F(n + 1) - 995/2 F(n) F(n + 1) - 1207/3 F(n) F(n + 1) 3 6 9 - 38315/6 F(n) F(n + 1) - 3205 F(n + 1) - 8 F(n) + 467/6 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, 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(2 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 15 2 3 81 3 G(n) = 9/22 F(n + 1) + -- F(n) - 9/22 F(n + 1) - -- F(n) 22 22 1395 6 3215 2 5 1685 4 3 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 22 22 11 435 2 525 3 4 - 6/11 F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 22 22 691 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, 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(2 j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 4 2 4 2 G(n) = 1/4 F(n) F(n + 1) - 1/6 + 1/6 F(n + 1) - 1/4 F(n) F(n + 1) 4 4 5 + 250/3 F(n) F(n + 1) + 1/2 F(n) F(n + 1) - 1/6 F(n) F(n + 1) 3 3 3 3 5 - 104/3 F(n) F(n + 1) - 1/2 F(n) F(n + 1) + 850/3 F(n) F(n + 1) 5 3 6 1531 2 2 - 675/2 F(n) F(n + 1) + 1/4 F(n) + ---- F(n) F(n + 1) 12 1525 2 6 59 4 - ---- F(n) F(n + 1) + -- 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, 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 ----- \ G(n) = ) F(j) F(2 j) F(2 n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 63 9 175 12313 3 6 G(n) = --- F(n + 1) + --- F(n) F(n + 1) + ----- F(n) F(n + 1) 638 319 319 2277013 2 7 467487 8 32097 4 - ------- F(n) F(n + 1) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 638 319 22 49035 4 77690 3 2 18121 2 3 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 638 319 29 119 2 2433 63 2 1960663 4 5 - --- F(n) - ---- F(n) - --- F(n + 1) + ------- F(n) F(n + 1) 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, 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(2 j) F(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 397 5 3 45 6 267 5 2 G(n) = ---- F(n) F(n + 1) + -- F(n) F(n + 1) + --- F(n) F(n + 1) 11 44 44 3 13 7 267 3 4 70 4 + 7/11 F(n) F(n + 1) - -- F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) 22 44 11 3 21 7 6 + 9/22 F(n + 1) - 9/22 - -- F(n) + 3/22 F(n) F(n + 1) 22 1107 6 2 771 7 19 3 5 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) + -- F(n) F(n + 1) 22 22 22 21 4 3 146 4 4 81 2 + -- F(n) F(n + 1) + --- F(n) F(n + 1) - -- F(n) F(n + 1) 11 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, 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 ----- \ G(n) = ) F(j) F(2 j) F(3 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 3 3 6 G(n) = -1/4 F(n) F(n + 1) - 2/3 F(n) F(n + 1) - 17/2 F(n) F(n + 1) 4 7 2 8 + 317/6 F(n) F(n + 1) - 217 F(n) F(n + 1) - 27/4 F(n) F(n + 1) 6 3 6 4 5 + 590/3 F(n) F(n + 1) - 2/3 F(n) F(n + 1) - 59/6 F(n) F(n + 1) 5 2 9 9 7 + 1/3 F(n) F(n + 1) - 1/6 F(n + 1) - 23/3 F(n) - 1/3 F(n) 3 2 8 + 1/6 F(n + 1) + 3/4 F(n) F(n + 1) + 3/4 F(n) F(n + 1) 3 4 4 3 - 1/3 F(n) F(n + 1) + 2/3 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, 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 ----- \ G(n) = ) F(j) F(2 j) F(4 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 27 4 63487 5 106 1829 G(n) = --- F(n + 1) + ----- F(n + 1) - --- F(n) + ---- F(n + 1) 22 22 11 22 65325 9 13715 4 63 2 2 - ----- F(n + 1) - ----- F(n) F(n + 1) + -- F(n) F(n + 1) 22 11 11 7548 2 3 4486 3 2 24 3 + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) + -- F(n) F(n + 1) 11 11 11 11075 2 7 64700 3 6 8 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + 14725/2 F(n) F(n + 1) 22 11 78 3 18 - -- 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, 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 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6360 2 3210 2 5 1404 3 G(n) = ---- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) 319 319 319 50007 3 128 120710 6 108800 3 4 + ----- F(n + 1) + --- + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 319 319 319 319 50135 7 20076 2 - ----- 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, 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 ----- \ G(n) = ) F(j) F(3 j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 1913 2 6 975 7 G(n) = -31 F(n + 1) + ---- F(n) F(n + 1) - --- F(n) F(n + 1) 21 14 373 2 2 3509 3 2119 7 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 42 42 21 545 8 505 8 797 - --- F(n) + --- F(n + 1) + --- 42 42 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, 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 ----- \ G(n) = ) F(j) F(3 j) F(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3344445 5 391515 2 3 5455 47095 G(n) = ------- F(n + 1) + ------ F(n) F(n + 1) - ---- F(n) + ----- F(n + 1) 1102 551 551 551 55 180975 9 6841125 3 6 - --- - ------ F(n + 1) - ------- F(n) F(n + 1) 551 58 1102 1437745 4 8514275 8 273050 2 7 - ------- F(n) F(n + 1) + ------- F(n) F(n + 1) - ------ F(n) F(n + 1) 1102 1102 551 461425 3 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, 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 ----- \ G(n) = ) F(j) F(3 j) F(6 j) / ----- j = 0 then we have the following closed-form expression for G(n) 105703 7 3 32335 5 5 7921 3 7 G(n) = ------- F(n) F(n + 1) - ----- F(n) F(n + 1) + ---- F(n) F(n + 1) 154 154 462 130646 6 4 530 2 8 16 2 32 + ------ F(n) F(n + 1) + --- F(n) F(n + 1) + -- F(n + 1) - --- 231 231 77 231 6 76192 9 16 6 - 1446/7 F(n) + ----- F(n) F(n + 1) - --- F(n + 1) 231 231 5333 3 3 51446 10 - ---- F(n) F(n + 1) + ----- F(n) 154 231 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, 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 ----- \ G(n) = ) F(j) F(3 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 5 2 3 G(n) = 9/22 F(n) F(n + 1) + 7/22 F(n) F(n + 1) - 4/11 F(n) F(n + 1) 181 2 4 3 2 218 3 3 + --- F(n) F(n + 1) - 9/22 F(n) F(n + 1) - --- F(n) F(n + 1) 22 11 4 102 5 2 + 4/11 F(n) F(n + 1) + --- F(n) F(n + 1) + 1/11 F(n + 1) 11 5 223 6 5 92 2 - 1/11 F(n + 1) + --- F(n) + 5/22 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, 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 ----- \ G(n) = ) F(j) F(3 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 3 2 4 G(n) = 1/6 F(n) F(n + 1) + 1/12 F(n) F(n + 1) - 1/6 F(n) F(n + 1) 2 4 2 + 1/4 F(n) F(n + 1) - 1/12 F(n) F(n + 1) + 125/6 F(n) F(n + 1) 805 5 2 2 5 395 4 3 - --- F(n) F(n + 1) - 75/4 F(n) F(n + 1) + --- F(n) F(n + 1) 12 12 3 4 47 3 3 + 215/6 F(n) F(n + 1) - -- F(n) - 1/6 F(n + 1) + 1/6 F(n + 1) 12 5 - 1/12 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, 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 ----- \ G(n) = ) F(j) F(3 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 224061 4 41251 2 2 466075 3 5 1986 G(n) = ------ F(n + 1) + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) + ---- 319 319 319 319 15812 3 97167 3 226175 8 - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) - ------ F(n + 1) 319 319 319 553925 7 16050 2 6 72 128 + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) - --- F(n) + --- F(n + 1) 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, 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 ----- \ G(n) = ) F(j) F(3 j) F(n + 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 17863 2 5356011 5 6833537 6 G(n) = ----- F(n) - ------- F(n) F(n + 1) + ------- F(n + 1) 1102 1102 551 171225 2 8 1927919 2 4 - ------ F(n) F(n + 1) + ------- F(n) F(n + 1) 58 551 1610819 3 3 14047925 3 7 81498 - ------- F(n) F(n + 1) - -------- F(n) F(n + 1) - ----- F(n) F(n + 1) 551 551 551 18108150 9 7253850 10 231 55 + -------- F(n) F(n + 1) - ------- F(n + 1) - ---- F(n) - --- F(n + 1) 551 551 1102 551 420368 2 + ------ 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, 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 ----- \ G(n) = ) F(j) F(3 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 2 2 25 2 4 G(n) = 1/22 F(n + 1) - 7/22 F(n) - 7/22 F(n + 1) - -- F(n) F(n + 1) 22 435 3 4 25 4 2 585 4 3 + --- F(n) F(n + 1) + -- F(n) F(n + 1) + --- F(n) F(n + 1) 11 22 22 1393 5 2 81 3 3 6 - ---- F(n) F(n + 1) - -- F(n) + 1/22 F(n + 1) + 5/22 F(n + 1) 22 22 215 2 5 5 223 2 - --- F(n) F(n + 1) + 5/11 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, 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 ----- \ 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) 2 2 749 6 2 G(n) = 1/4 F(n) - 1/6 + 1/6 F(n + 1) + --- F(n) F(n + 1) 15 5 3 71 8 167 3 167 7 - 66 F(n) F(n + 1) + -- F(n) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 12 10 10 7 2 2 - 1/6 F(n) F(n + 1) - 155/6 F(n) F(n + 1) - 8/5 F(n) F(n + 1) 2 6 + 75/2 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, 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 ----- \ G(n) = ) F(j) F(3 j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 128 200 2 144 5 467298 4 G(n) = ---- F(n + 1) + --- F(n) - --- F(n) F(n + 1) - ------ F(n) F(n + 1) 319 319 319 319 980300 4 5 12250 3 6 75001 3 2 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 319 1138475 2 7 203112 2 3 467550 8 - ------- F(n) F(n + 1) + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 319 319 319 288 4 2 144 3 3 288 2 4 - --- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 3390 5 144 5 20896 4 128 2 - ---- F(n) - --- F(n) F(n + 1) + ----- F(n) F(n + 1) + --- F(n + 1) 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, 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 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 815 3 6849 3 21 2 G(n) = 1/11 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) - -- F(n) F(n + 1) 22 22 22 2715 2 2 4 4 3425 7 + ---- F(n) F(n + 1) + 1375/2 F(n) F(n + 1) + ---- F(n) F(n + 1) 22 11 16375 2 6 3 3 - ----- F(n) F(n + 1) + 6/11 F(n) - 1/11 F(n + 1) 22 775 3 5 59 4 15 2 - --- F(n) F(n + 1) + -- F(n) + -- 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, 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 j) F(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 4 G(n) = 1/6 F(n + 1) - 29/3 F(n) + 343/4 F(n + 1) - 1383 F(n) F(n + 1) 2 2 3 6025 2 7 + 3/4 F(n) F(n + 1) + 2227/3 F(n) F(n + 1) - ---- F(n) F(n + 1) 12 3 2 3 6 97975 8 - 429 F(n) F(n + 1) - 19750/3 F(n) F(n + 1) + ----- F(n) F(n + 1) 12 2 3 9 38569 5 - 1/4 F(n) F(n + 1) - 1/3 F(n) - 3300 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, 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 j) F(3 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 128 3 272 3 5376 2 4132800 10 G(n) = --- F(n + 1) - --- F(n) + ---- F(n) - ------- F(n + 1) 319 319 319 319 216 2 7997400 3 7 8706534 3 3 - --- F(n) F(n + 1) - ------- F(n) F(n + 1) - ------- F(n) F(n + 1) 319 319 319 934475 2 8 4990153 2 4 600 2 - ------ F(n) F(n + 1) + ------- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 10319325 9 569123 5 48154 + -------- F(n) F(n + 1) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 319 29 319 3888756 4 2 4132672 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, 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 ----- \ G(n) = ) F(j) F(3 j) F(4 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 1039 5 115 517 9 172 6 2 G(n) = ---- F(n + 1) + --- - --- F(n) - --- F(n) F(n + 1) 165 66 40 55 25019 6 3 215 7 53657 7 2 + ----- F(n) F(n + 1) - --- F(n) F(n + 1) - ----- F(n) F(n + 1) 165 33 660 12257 8 817 7 553 8 + ----- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 330 165 264 86 8 86 8 9851 5 4 821 4 - -- F(n + 1) - -- F(n) - ---- F(n) F(n + 1) - --- F(n) F(n + 1) 33 33 120 44 256 3 977 2 2 17 4 1667 + --- F(n) F(n + 1) + --- F(n) F(n + 1) + -- F(n + 1) + ---- F(n) 55 330 22 440 1024 - ---- F(n + 1) 165 Proof: Both 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 j) F(4 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 7 6 185 2 G(n) = -27075 F(n) F(n + 1) + 79033/6 F(n + 1) + --- F(n) 12 167425 10 3 5 - ------ F(n + 1) - 5/2 F(n) F(n + 1) + 7/12 - 15586/3 F(n) F(n + 1) 12 3 3 3 9 + 5/6 F(n) F(n + 1) - 6037/2 F(n) F(n + 1) + 104425/3 F(n) F(n + 1) 36725 2 8 44087 2 4 25 2 2 - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) + -- F(n) F(n + 1) 12 12 12 4 2 - 5/12 F(n + 1) + 3119/4 F(n + 1) - 893/6 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, 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 j) F(5 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 10 31 19179 3 7 157 4 G(n) = 1/11 F(n + 1) + -- F(n) + ----- F(n) F(n + 1) - --- F(n) F(n + 1) 22 22 22 8219 4 2 4 5 137351 4 6 + ---- F(n) F(n + 1) - 1/11 F(n) F(n + 1) + ------ F(n) F(n + 1) 11 11 2 3 57895 2 4 2 7 - 4/11 F(n) F(n + 1) + ----- F(n) F(n + 1) + 1/11 F(n) F(n + 1) 22 133887 5 321 2 3 6 - ------ F(n) F(n + 1) + --- F(n) - 2/11 F(n) F(n + 1) 22 22 4 338477 2 8 151 3 2 - 3/11 F(n) F(n + 1) - ------ F(n) F(n + 1) + --- F(n) F(n + 1) 22 22 3 3 8 137021 9 - 1366 F(n) F(n + 1) + 2/11 F(n) F(n + 1) + ------ F(n) F(n + 1) 22 3151 9 - ---- F(n) F(n + 1) - 1/11 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, 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 ----- \ 2 G(n) = ) F(j) F(4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4841 2759 5 441 4429 4 G(n) = ----- F(n) - ---- F(n) + ---- + ---- F(n) F(n + 1) 1102 551 1102 551 126857 5 4 441 9 63 8 - ------ F(n) F(n + 1) - ---- F(n + 1) - -- F(n) F(n + 1) 1102 1102 29 36605 4 5 41936 6 3 53640 7 2 + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 551 551 551 40475 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, 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(4 j) F(5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 15 10 229475 4 6 1140 2 G(n) = --- F(n + 1) - ------ F(n) F(n + 1) + ---- F(n) 77 77 77 118535 237610 5 17000 9 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 77 77 11 216395 3 3 694175 3 7 630255 5 5 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 77 77 77 15 273380 4 2 107355 5 + -- + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 77 77 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, 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(4 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 209 7 6 G(n) = 1/12 F(n) - 1/6 F(n + 1) + --- F(n + 1) + 85/6 F(n) F(n + 1) 60 37 7 4 3 199 3 1457 5 2 - -- F(n) + 61/3 F(n) F(n + 1) - --- F(n + 1) - ---- F(n) F(n + 1) 12 60 60 2 95 6 + 7/10 F(n) F(n + 1) - -- 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, 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(4 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 226175 8 1986 82821 2 2 15493 3 G(n) = ------- F(n + 1) + ---- + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 638 319 448441 4 16050 2 6 195291 3 + ------ F(n + 1) - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 638 319 638 553925 7 466075 3 5 72 63 + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) - --- F(n) - --- 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, 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 ----- \ G(n) = ) F(j) F(4 j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 441 9 74987432 3 3 7497 4 G(n) = ---- F(n + 1) - -------- F(n) F(n + 1) + ----- F(n) F(n + 1) 1102 24795 20938 87788279 8 2 60929342 9 + -------- F(n) F(n + 1) + -------- F(n) F(n + 1) 24795 24795 10584 8 28566902 9 10143 2 3 - ----- F(n) F(n + 1) + -------- F(n) F(n + 1) - ----- F(n) F(n + 1) 10469 24795 20938 1379543 2 4 1323 2 7 - ------- F(n) F(n + 1) + ----- F(n) F(n + 1) 1653 10469 53450684 2 8 18081 3 2 31311 3 6 - -------- F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 24795 20938 41876 1323 4 14608973 1546246 5 - ---- F(n) F(n + 1) - -------- F(n) F(n + 1) - ------- F(n) F(n + 1) 2204 24795 2755 441 10 8337 8318899 10 441 9 - ---- F(n + 1) - ----- F(n) + ------- F(n) - ----- F(n) 1102 41876 16530 41876 4033742 2 - ------- F(n) 8265 Proof: Both 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(4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 7 1553 2 2 G(n) = -1/4 F(n) + 675/2 F(n) F(n + 1) + ---- F(n) F(n + 1) 12 9625 2 6 3 3 - ---- F(n) F(n + 1) - 75/2 F(n) F(n + 1) - 675/2 F(n) F(n + 1) 12 4 4 73 4 + 2275/3 F(n) F(n + 1) + 1/6 F(n) F(n + 1) + -- F(n) + 1/6 12 3 5 2 - 325/6 F(n) 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, 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(4 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 864 5 5823 9 63 2 72 6 63 G(n) = ---- F(n) F(n + 1) - ---- F(n) - --- F(n) + --- F(n) + --- F(n + 1) 319 638 638 319 638 360 6 51023 8 171123 2 3 + --- F(n + 1) - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 319 638 1276 40579 3 6 568203 4 869663 8 - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 145 6380 6380 720 3 3 257253 3 2 376715 2 7 + --- F(n) F(n + 1) + ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 319 3190 1276 25604 4 27 2 + ----- F(n) F(n + 1) - -- 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, 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(4 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 3 6 G(n) = 1/2 F(n) F(n + 1) + 1/3 F(n) + 1/4 F(n) F(n + 1) 3 2 4 9 - 1205/6 F(n) F(n + 1) + 193/4 F(n) F(n + 1) - 28/3 F(n) 16531 8 2 2 5 + ----- F(n) F(n + 1) - 3/4 F(n) F(n + 1) - 1/2 F(n) F(n + 1) 12 40027 2 7 3 4 3 6 - ----- F(n) F(n + 1) - 1/4 F(n) F(n + 1) - 89/3 F(n) F(n + 1) 12 4 5 4 5 2 + 5891/2 F(n) F(n + 1) - 4135/3 F(n) F(n + 1) + 1/4 F(n) F(n + 1) 6991 2 3 3 9 + ---- F(n) F(n + 1) - 1/6 F(n + 1) + 1/6 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, 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(4 j) F(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 433 2 8 40771 3 3 G(n) = 7/58 F(n) F(n + 1) + --- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 1276 23587 3 7 103053 5 37173 5 5 + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 1276 638 1276 535063 7 3 337877 8 2 4545 9 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) + ---- F(n) F(n + 1) 1276 638 58 243 6 1215 5 2 63 10 9795 10 + --- F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n + 1) + ---- F(n) 638 1276 638 638 1215 3 4 225 3 63 3 243 6 - ---- F(n) F(n + 1) - --- F(n) - --- F(n + 1) + ---- F(n) F(n + 1) 1276 638 638 1276 1107 2 - ---- 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, 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(4 j) F(4 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 289627 2 25 8 105633 6 17 25 8 G(n) = ------- F(n + 1) - -- F(n + 1) + ------ F(n + 1) + -- - -- F(n) 1860 21 620 28 21 125569 2 125 7 3280 9 - ------ F(n) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 930 42 93 15 2 2 2 6 6821 + -- F(n) F(n + 1) - 10/7 F(n) F(n + 1) + ---- F(n) F(n + 1) 28 930 24634 7 3 8 2 83219 9 - ----- F(n) F(n + 1) + 615 F(n) F(n + 1) + ----- F(n) F(n + 1) 31 186 20 3 100917 5 95 7 + -- F(n) F(n + 1) - ------ F(n) F(n + 1) - -- F(n) F(n + 1) 21 310 42 280123 10 13481 10 4 + ------ F(n) - ----- F(n + 1) + 5/12 F(n + 1) 1860 930 Proof: Both 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(5 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 16880 5 3 215 7 G(n) = ------ F(n) F(n + 1) + --- F(n) F(n + 1) + 5/638 - 5/638 F(n + 1) 319 638 15 5 34985 4 4 57975 6 2 + --- F(n) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 319 1276 1276 7965 7 15 4 3535 2 2 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 319 319 1276 30 2 3 15 3 2 135 2 6 - --- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 319 319 44 30 4 3155 8 + --- 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, 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(5 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 3 5 G(n) = 5/6 F(n) (-12 + 1827 F(n + 1) + 3846 F(n) F(n + 1) 2 6 2 2 7 + 29 F(n) F(n + 1) - 302 F(n) F(n + 1) - 4443 F(n) F(n + 1) 3 4 3 + 767 F(n) F(n + 1) - 1815 F(n + 1) + 103 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, 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(5 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 20 5 4 55 9 1676 9 2476 5 G(n) = --- F(n) F(n + 1) - --- F(n + 1) - ---- F(n) + ---- F(n) 29 551 2755 2755 324 4 29455 5 56 8 + --- F(n) F(n + 1) + ----- F(n) F(n + 1) + --- F(n) F(n + 1) 551 4408 551 29085 9 102 2 3 1488 4 - ----- F(n) F(n + 1) - --- F(n) F(n + 1) + ---- F(n) F(n + 1) 8816 551 2755 37 8 35455 8 2 366213 9 - -- F(n) F(n + 1) + ----- F(n) F(n + 1) - ------ F(n) F(n + 1) 29 76 8816 358919 4 2 2537 4 5 135891 4 6 - ------ F(n) F(n + 1) + ---- F(n) F(n + 1) + ------ F(n) F(n + 1) 2204 2755 551 179981 5 4254625 5 5 1125 - ------ F(n) F(n + 1) - ------- F(n) F(n + 1) - ---- F(n) F(n + 1) 4408 8816 304 55 10 8105 10 + --- F(n + 1) + ---- F(n) 551 551 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, 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 ----- \ G(n) = ) F(j) F(5 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 1948350 3 6 2428150 8 2 G(n) = -------- F(n) F(n + 1) + ------- F(n) F(n + 1) - 5/638 F(n + 1) 319 319 980300 9 410035 4 953175 5 - ------ F(n + 1) - ------ F(n) F(n + 1) + ------ F(n + 1) 319 319 319 132815 3 2 35 2 30 - ------ F(n) F(n + 1) - --- F(n) + --- F(n) F(n + 1) 319 638 319 224385 2 3 158175 2 7 6345 + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) - ---- F(n) 319 319 638 54255 + ----- 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, 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(5 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 10 7 3 6 G(n) = 5/8 F(n + 1) - 2935/6 F(n) F(n + 1) - 5/4 F(n + 1) 4 6 173 4 2 4739 8 2 + 2965/8 F(n) F(n + 1) + --- F(n) F(n + 1) + ---- F(n) F(n + 1) 12 24 3 7 3 3 9 - 521/3 F(n) F(n + 1) + 907/6 F(n) F(n + 1) - 515/6 F(n) F(n + 1) 10 2 + 15 F(n) + 5/8 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, 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(5 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 525 6 54245 4 6 84015 6 4 G(n) = ---- F(n) F(n + 1) + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 638 1276 1276 143580 7 3 705115 8 2 56555 9 - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 319 1276 638 45 5 2 15 2 5 155 6 + --- F(n) F(n + 1) - -- F(n) F(n + 1) + ---- F(n + 1) 319 29 1276 2 25 3 9505 10 105 7 25 3 - 5/44 F(n + 1) + -- F(n) + ---- F(n) - --- F(n) - --- F(n + 1) 58 638 319 319 375 4 3 3195 3 3 55435 5 + --- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 319 1275 2 4 45 7 + ---- F(n) F(n + 1) + --- F(n + 1) 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, 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 ----- \ G(n) = ) F(j) F(6 j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 9 8 G(n) = -3205 F(n + 1) - 8 F(n) + 3205 F(n + 1) + 15865/2 F(n) F(n + 1) 2 3 4 + 23089/6 F(n) F(n + 1) - 9380/3 F(n) F(n + 1) 3 6 3 2 - 38315/6 F(n) F(n + 1) - 39949/6 F(n) F(n + 1) 2 7 4 - 995/2 F(n) F(n + 1) + 9791/2 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, 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(6 j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 39604842 9 97420 2 54 10 G(n) = -------- F(n) F(n + 1) + ----- F(n) + ---- F(n + 1) 6061 6061 6061 38711942 5 893723 - -------- F(n) F(n + 1) - ------ F(n) F(n + 1) 6061 6061 195370833 2 8 108 8 108 4 - --------- F(n) F(n + 1) + ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 12122 6061 6061 108 3 2 8474770 3 3 54 2 3 - ---- F(n) F(n + 1) - ------- F(n) F(n + 1) + ---- F(n) F(n + 1) 6061 6061 6061 16600142 2 4 54 2 7 54 4 + -------- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 6061 6061 6061 9227709 4 2 54 4 5 + ------- F(n) F(n + 1) - ---- F(n) F(n + 1) 12122 6061 79792404 4 6 5057633 3 7 + -------- F(n) F(n + 1) + ------- F(n) F(n + 1) 6061 6061 108 3 6 444 54 9 - ---- F(n) F(n + 1) - ---- F(n) - ---- 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, 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(6 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 G(n) = 1/1230 F(n) (947 F(n + 1) + 140327 F(n) F(n + 1) 6 3 9 4 - 873407 F(n) F(n + 1) - 136313 F(n + 1) + 66221 F(n) F(n + 1) 8 5 7 2 + 272010 F(n) F(n + 1) + 135366 F(n + 1) + 375169 F(n) F(n + 1) 9 + 80931 F(n) - 61251 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, 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 ----- \ 2 G(n) = ) F(j) F(n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 G(n) = -1/2 F(n + 1) + 1/2 F(n + 1) + 3/2 F(n) - F(n) F(n + 1) 2 3 3 2 4 5 + 1/2 F(n) F(n + 1) - 1/2 F(n) F(n + 1) + 2 F(n) F(n + 1) - 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, 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(n + j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 3 2 2 3 G(n) = -F(n) (3 F(n) F(n + 1) - 13 F(n) F(n + 1) + 12 F(n) F(n + 1) - 2) Proof: Both 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(n + j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 691 2 2 2 35 3 G(n) = ---- F(n) F(n + 1) + 2/11 F(n) - 1/11 F(n + 1) - -- F(n) 11 11 3 212 2 525 3 4 + 1/11 F(n + 1) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 11 22 1395 6 3215 2 5 - 1/22 F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 22 22 1685 4 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, 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(n + j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 5 7 G(n) = -328 F(n) F(n + 1) - 9425/6 F(n) F(n + 1) + 11125/6 F(n) F(n + 1) 2 6 3 8 59 - 175/4 F(n) F(n + 1) - 46 F(n) F(n + 1) - 2275/3 F(n + 1) + -- 12 9043 4 2 2 2 + ---- F(n + 1) + 805/6 F(n) F(n + 1) + 1/6 F(n) F(n + 1) + 1/12 F(n) 12 2 - 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, 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(n + j) F(n + 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 9 32059 4 G(n) = -5/638 F(n + 1) + 5/638 F(n + 1) - ----- F(n) F(n + 1) 22 12255 3 6 1960605 4 5 27 + ----- F(n) F(n + 1) + ------- F(n) F(n + 1) - --- F(n) F(n + 1) 319 638 638 154249 3 2 49325 4 198258 2 3 - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 638 638 319 2276955 2 7 467545 8 23 2 5127 - ------- F(n) F(n + 1) + ------ F(n) F(n + 1) + --- F(n) - ---- F(n) 638 319 638 638 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, 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(n + j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 3 3 2 G(n) = -12 F(n) F(n + 1) + 83 F(n) F(n + 1) - 76 F(n) F(n + 1) + 5/2 F(n) 2 2 6 3 + 34 F(n + 1) + F(n) F(n + 1) - 69/2 F(n + 1) - 1/2 F(n) 3 2 2 4 + 1/2 F(n + 1) - 3/2 F(n) F(n + 1) + 7/2 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, 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 ----- \ G(n) = ) F(j) F(n + j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 12 7 887 4 4 491 5 3 G(n) = -7/44 F(n + 1) + -- F(n) + --- F(n) F(n + 1) - --- F(n) F(n + 1) 11 44 11 1019 6 2 541 7 111 8 + ---- F(n) F(n + 1) - --- F(n) F(n + 1) - 1/11 + --- F(n) 22 22 22 5 2 59 2 5 30 6 - 7/22 F(n) F(n + 1) + -- F(n) F(n + 1) + -- F(n) F(n + 1) 44 11 15 8 3 23 3 113 2 6 + -- F(n + 1) + 3/44 F(n + 1) - -- F(n) - --- F(n) F(n + 1) 44 22 44 155 4 3 4 - --- F(n) F(n + 1) - 7/44 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, 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 ----- \ G(n) = ) F(j) F(n + j) F(2 n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 7919 2 3 2 G(n) = 6257/4 F(n) F(n + 1) + ---- F(n) F(n + 1) - 1/12 F(n) F(n + 1) 12 3 2 2 7 3 6 - 507/2 F(n) F(n + 1) - 15209/4 F(n) F(n + 1) + 17 F(n) F(n + 1) 4 5 9 2 + 19801/6 F(n) F(n + 1) + 1/6 F(n + 1) + 1/4 F(n) F(n + 1) - 8 F(n) 4 4 3 + 317/4 F(n) F(n + 1) - 1557 F(n) 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, 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(n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 3 2 2 3 G(n) = -7 F(n) + 1/2 - F(n) F(n + 1) + 7/2 F(n) F(n + 1) - 4 F(n) F(n + 1) 3 4 2 3 3 - 25/2 F(n) F(n + 1) + 5/2 F(n) F(n + 1) + 7/2 F(n) - 1/2 F(n + 1) 2 6 2 5 - 9 F(n) F(n + 1) + 35/2 F(n) F(n + 1) + 13/2 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, 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(n + j) F(3 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 79413 179 9 54973 9 4 G(n) = ------ F(n + 1) - --- F(n) + ----- F(n + 1) + 3/22 - 5/22 F(n + 1) 836 22 418 27 2 2 104897 2 3 1029 4 + -- F(n) F(n + 1) - ------ F(n) F(n + 1) - ---- F(n) F(n + 1) 22 418 19 244229 2 7 13 3 10599 3 2 + ------ F(n) F(n + 1) - -- F(n) F(n + 1) + ----- F(n) F(n + 1) 836 22 38 1603 5 3 83123 8 - ---- F(n + 1) - 7/22 F(n) F(n + 1) - ----- F(n) F(n + 1) 44 209 116691 8 + ------ F(n) F(n + 1) 836 Proof: Both 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(n + j) F(4 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 4 6 2 G(n) = -23/2 F(n) F(n + 1) + 21/2 F(n) F(n + 1) + 261/2 F(n) F(n + 1) 3 2 6 3 5 - 36 F(n) F(n + 1) - 7 F(n) F(n + 1) + 64 F(n) F(n + 1) 4 4 5 3 4 4 - 93 F(n) F(n + 1) - 64 F(n) F(n + 1) + 15/2 F(n) + F(n + 1) - 3/2 29 19 5 5 - -- F(n) - 1/5 F(n + 1) + -- F(n) + 7/10 F(n + 1) 10 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, 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(n + j) F(5 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 8 G(n) = 1/2 F(n + 1) + 3/2 F(n) + 2877/2 F(n) F(n + 1) 4 4 5 4 - 2849/2 F(n) F(n + 1) + 2912 F(n) F(n + 1) + 89 F(n) F(n + 1) 3 6 3 3 3 2 + 323/2 F(n) F(n + 1) - 30 F(n) F(n + 1) - 261 F(n) F(n + 1) 2 7 2 3 4 2 - 3537 F(n) F(n + 1) + 634 F(n) F(n + 1) + 85/2 F(n) F(n + 1) 5 9 + 10 F(n) F(n + 1) - 1/2 F(n + 1) - 12 F(n) F(n + 1) 2 4 - 15 F(n) F(n + 1) - 19/2 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, 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 ----- \ 2 G(n) = ) F(j) F(n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 691 2 G(n) = -9/22 F(n + 1) + 9/22 F(n + 1) - --- F(n) F(n + 1) 11 525 3 4 3215 2 5 457 2 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) + --- F(n) F(n + 1) 22 22 22 1395 6 1685 4 3 + ---- F(n) F(n + 1) - 6/11 F(n) F(n + 1) + ---- F(n) F(n + 1) 22 11 81 3 2 - -- F(n) - 7/22 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, 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(n + 2 j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 5 2 77 4 G(n) = -1/6 F(n) F(n + 1) - 325/6 F(n) F(n + 1) - 1/6 - 1/4 F(n) + -- F(n) 12 2 3 7 + 1/6 F(n + 1) - 675/2 F(n) F(n + 1) + 675/2 F(n) F(n + 1) 3 9625 2 6 1549 2 2 - 223/6 F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 12 12 4 4 + 2275/3 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, 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(n + 2 j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 6461 54139 953262 5 63 2 G(n) = ----- F(n) + ----- F(n + 1) + ------ F(n + 1) - --- F(n + 1) 638 638 319 638 158175 2 7 1948350 3 6 - ------ F(n) F(n + 1) - ------- F(n) F(n + 1) 319 319 265253 3 2 2428150 8 81 2 - ------ F(n) F(n + 1) + ------- F(n) F(n + 1) + --- F(n) 638 319 638 31 448973 2 3 820621 4 + --- F(n) F(n + 1) + ------ F(n) F(n + 1) - ------ F(n) F(n + 1) 638 638 638 980300 9 - ------ 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, 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 ----- \ G(n) = ) F(j) F(n + 2 j) F(n + 5 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 4 G(n) = -1/6 F(n) (-7655 F(n + 1) + 47905 F(n + 1) - 33948 F(n) F(n + 1) 8 2 7 3 2 + 99225 F(n) F(n + 1) - 4975 F(n) F(n + 1) + 12471 F(n) F(n + 1) 3 6 4 9 - 81225 F(n) F(n + 1) + 8548 F(n) F(n + 1) - 96 F(n) - 40250 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, 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 ----- \ 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) 8 3 625 2 6 3 G(n) = -1375/2 F(n + 1) + 1/22 F(n) - --- F(n) F(n + 1) + 9/22 F(n + 1) 11 2 14985 4 2 - 9/11 F(n) F(n + 1) + ----- F(n + 1) + 1/22 F(n) F(n + 1) 22 18550 7 6569 3 31025 3 5 + ----- F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 11 22 22 2833 2 2 1073 3 131 + ---- F(n) F(n + 1) - ---- F(n) 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, 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(n + 2 j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 1429 4 5 257 127 G(n) = 5/12 F(n + 1) + ---- F(n) F(n + 1) - --- F(n + 1) - --- F(n) 24 120 60 2 5 7381 5 57 5 4325 8 - 1/3 F(n) F(n + 1) - ---- F(n) + -- F(n + 1) + ---- F(n) F(n + 1) 120 20 24 3625 7 2 6 1283 9 7 - ---- F(n) F(n + 1) + 2/3 F(n) F(n + 1) + ---- F(n) + 1/6 F(n) 24 24 7 1877 5 4 6 - 1/4 F(n + 1) - ---- F(n) F(n + 1) + 1/12 F(n) F(n + 1) 24 5 2 2 9 - 1/2 F(n) F(n + 1) - 5/12 F(n) F(n + 1) - 7/8 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, 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(n + 2 j) F(2 n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 206 4 3 147981 10 121 3 9659 2 G(n) = --- F(n) F(n + 1) + ------ F(n) - --- F(n) - ---- F(n) 319 2552 290 232 103 7 16 2 51 9 + ---- F(n) + ---- F(n) F(n + 1) - -- F(n) F(n + 1) 1595 1595 58 508229 6 4 26953 9 362113 8 2 + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 2552 319 1276 857753 7 3 26189 3 3 10883 2 8 - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 1595 290 12760 1236 2 5 283761 2 4 103 6 - ---- F(n) F(n + 1) + ------ F(n) F(n + 1) + --- F(n) F(n + 1) 1595 6380 319 63 3 63 10 - --- F(n + 1) + --- 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, 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(n + 2 j) F(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 14315 95045 4 785 4 G(n) = ----- F(n) - ----- F(n) F(n + 1) + --- F(n) F(n + 1) 209 418 11 18 3 69941 3 6 57643 8 - -- F(n) F(n + 1) - ----- F(n) F(n + 1) - ----- F(n) F(n + 1) 11 209 418 3 181471 2 7 46864 3 2 - 3/11 F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 418 209 7937 2 3 2 2 17 4 - ---- F(n) F(n + 1) + 3/2 F(n) F(n + 1) - -- F(n) + 9/22 418 22 9 16329 9 - 9/22 F(n + 1) - ----- F(n) 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, 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(n + 2 j) F(3 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 3 197 6 10 G(n) = 7/6 F(n) F(n + 1) + --- F(n) - 1/6 F(n + 1) + 1/6 12 7 8 301 9 - 1/30 F(n) F(n + 1) - 7/12 F(n) - --- F(n) F(n + 1) 96 139 7 42955 7 3 - --- F(n) F(n + 1) - 4/3 F(n) F(n + 1) - ----- F(n) F(n + 1) 32 32 11 6 2 6 4 5 5 + -- F(n) F(n + 1) + 4099/8 F(n) F(n + 1) + 4973/6 F(n) F(n + 1) 60 22769 4 2 4 4 18035 4 6 + ----- F(n) F(n + 1) - 1/4 F(n) F(n + 1) - ----- F(n) F(n + 1) 48 48 3653 5 2 2 133 5 - ---- F(n) F(n + 1) + 7/30 F(n) F(n + 1) + --- F(n) F(n + 1) 32 16 3 + 1/30 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, 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(n + 2 j) F(4 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 353 3 6 10 3033 10 17 G(n) = ---- F(n) F(n + 1) - 9/22 F(n + 1) - ---- F(n) - -- F(n) 44 22 44 97 4 5 474 4 6 1713 5 + -- F(n) F(n + 1) + --- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 11 44 54 5 4 2217 5 5 30 6 3 + -- F(n) F(n + 1) - ---- F(n) F(n + 1) - -- F(n) F(n + 1) 11 22 11 34 8 4750 8 2 3877 9 - -- F(n) F(n + 1) + ---- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 11 11 17 79 4 115 5 31 5 - -- F(n) F(n + 1) - -- F(n) F(n + 1) + --- F(n) F(n + 1) + -- F(n) 44 44 44 11 1688 6 89 9 9 41 3 3 + ---- F(n) - -- F(n) + 9/22 F(n + 1) + -- F(n) F(n + 1) 11 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, 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 ----- \ 2 G(n) = ) F(j) F(n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 16010 4 78966 3 2 26591 4 G(n) = ------ F(n) F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 11 319 319 144 467806 8 1138347 2 7 - --- F(n) F(n + 1) + ------ F(n) F(n + 1) - ------- F(n) F(n + 1) 319 319 319 11994 3 6 980172 4 5 128 2 + ----- F(n) F(n + 1) + ------ F(n) F(n + 1) + --- F(n + 1) 319 319 319 18005 2 3 2752 119 2 128 9 + ----- F(n) F(n + 1) - ---- F(n) - --- F(n) - --- F(n + 1) 29 319 319 319 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, 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 ----- \ G(n) = ) F(j) F(n + 3 j) F(n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 157475 9 157400 3 3 47675 G(n) = ------- F(n) F(n + 1) + ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 426 71 213 68485 5 362950 9 101950 2 8 - ----- F(n) F(n + 1) - ------ F(n) F(n + 1) + ------ F(n) F(n + 1) 71 213 71 217075 10 303655 2 2 14430 6 + ------ F(n + 1) - ------ F(n + 1) + 15 F(n) + ----- F(n + 1) 426 426 71 121185 2 4 - ------ F(n) F(n + 1) 142 Proof: Both 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 ----- \ 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) 1565 6 2 431 7 1433 3 G(n) = ---- F(n) F(n + 1) - --- F(n) F(n + 1) - ---- F(n) F(n + 1) 11 22 22 1363 2 6 3 4 5 2 - ---- F(n) F(n + 1) + 5/44 F(n) F(n + 1) + 3/44 F(n) F(n + 1) 22 5 3 29 3 3 195 8 + 27/2 F(n) F(n + 1) - -- F(n) - 2/11 F(n + 1) - --- F(n) 44 22 8 338 4 337 2 5 + 31/2 F(n + 1) - --- F(n + 1) + --- - 9/22 F(n) F(n + 1) 11 22 6 7 7 + 5/11 F(n) F(n + 1) + 9/44 F(n) + 1/11 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, 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 ----- \ 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) 3 4 3 3 4 G(n) = 1/6 F(n + 1) - 1/2 F(n) F(n + 1) + 1/4 F(n) F(n + 1) 2 5 8 5 3 + 1/2 F(n) F(n + 1) + 3/4 F(n) F(n + 1) - 55/6 F(n) + 1/6 F(n) 6 2 8111 2 3 - 1/4 F(n) F(n + 1) - 1/4 F(n) F(n + 1) + ---- F(n) F(n + 1) 12 4 5 5 4 8083 2 7 + 343/2 F(n) F(n + 1) - 9385/6 F(n) F(n + 1) - ---- F(n) F(n + 1) 12 3 2 3 6 803 4 - 745/3 F(n) F(n + 1) + 3161/2 F(n) F(n + 1) + --- F(n) F(n + 1) 12 5 2 9 - 1/4 F(n) 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, 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 ----- \ G(n) = ) F(j) F(n + 3 j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 102588 4 6 38 4 3 95 3 4 G(n) = ------ F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 319 319 319 26824 4 2 26969 9 306011 8 2 + ----- F(n) F(n + 1) - ----- F(n) F(n + 1) + ------ F(n) F(n + 1) 1595 319 1595 144487 7 3 19 5 2 20937 3 7 - ------ F(n) F(n + 1) - -- F(n) F(n + 1) - ----- F(n) F(n + 1) 319 55 145 219157 3 3 368 2 38 7 678 3 + ------ F(n) F(n + 1) - --- F(n + 1) - ---- F(n + 1) + ---- F(n + 1) 1595 319 1595 1595 28 3 4738 10 19 7 318 6 78 10 + --- F(n) + ---- F(n) + --- F(n) + --- F(n + 1) - --- F(n + 1) 319 319 319 319 319 1004 2 - ---- 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, 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 ----- \ 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 15555 4 31321 8 G(n) = 1/11 F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 11 22 2 2 6672 2 3 38201 2 7 - 9/11 F(n) F(n + 1) + ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 11 11 2690 3 2 4 4 2 6 - ---- F(n) F(n + 1) + 7/22 F(n) F(n + 1) - 7/22 F(n) F(n + 1) 11 7 3 65327 4 5 - 7/11 F(n) F(n + 1) - 9/22 F(n) F(n + 1) + ----- F(n) F(n + 1) 22 931 4 3 6 106 + --- F(n) F(n + 1) + 57 F(n) F(n + 1) - --- F(n) 11 11 3 5 23 4 8 + 7/11 F(n) F(n + 1) - 9/22 + -- F(n) + 7/22 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, 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 ----- \ G(n) = ) F(j) F(n + 3 j) F(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 2 G(n) = F(n) F(n + 1) - 442/3 F(n) F(n + 1) + 3111/4 F(n + 1) 167425 10 36725 2 8 44105 2 4 - ------ F(n + 1) - ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 12 12 12 2 2 9 5 - 5/12 F(n) F(n + 1) + 104425/3 F(n) F(n + 1) - 31205/6 F(n) F(n + 1) 3 3 7 3 3 - 7/6 F(n) F(n + 1) - 27075 F(n) F(n + 1) - 3015 F(n) F(n + 1) 6 185 2 4 + 79045/6 F(n + 1) + --- F(n) + 7/12 F(n + 1) - 5/12 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, 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 ----- \ G(n) = ) F(j) F(n + 3 j) F(4 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 5339 8 2 589 9 G(n) = -1/2 F(n) F(n + 1) + ---- F(n) F(n + 1) + --- F(n) F(n + 1) 11 22 3393 5 252 5 5 157 2 3 - ---- F(n) F(n + 1) - --- F(n) F(n + 1) - --- F(n) F(n + 1) 22 11 110 403 2 4 103 2 7 195 2 8 - --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 22 110 22 102 3 2 7499 7 3 9 - --- F(n) F(n + 1) - ---- F(n) F(n + 1) - 3/11 F(n) F(n + 1) 55 22 9 12 9 53 10 10 124 2 - 1/11 F(n + 1) - -- F(n) + -- F(n) + 1/11 F(n + 1) + --- F(n) 11 11 11 8 317 7 2 51 5 4 + 9/11 F(n) F(n + 1) + --- F(n) F(n + 1) - -- F(n) F(n + 1) 55 11 37 6 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, 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 ----- \ G(n) = ) F(j) F(n + 4 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 29 6 2 7 G(n) = --- F(n) F(n + 1) + 23/8 F(n) F(n + 1) + 5/24 F(n + 1) 60 5 7 7 17 2 - 1/24 F(n + 1) - 1/5 F(n) - 1/60 F(n + 1) + -- F(n) F(n + 1) 60 9 3 3 6 3 - 10 F(n) + 1/5 F(n) - 3/20 F(n + 1) + 1145/6 F(n) F(n + 1) 3 4 203 3 6 655 4 - 1/30 F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 12 12 4 3 3 2 7 2 + 2/5 F(n) F(n + 1) - 1/3 F(n) F(n + 1) - 213 F(n) F(n + 1) 197 8 - --- 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, 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 ----- \ G(n) = ) F(j) F(n + 4 j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 140960 7 3 98 2 63 3 G(n) = ------- F(n) F(n + 1) + --- F(n) F(n + 1) - --- F(n + 1) 319 319 638 1409 2 235 6 47 7 3329 10 + ---- F(n) + --- F(n) F(n + 1) + --- F(n) + ---- F(n) 319 638 319 319 113 2 868 2 4 47 2 5 - --- F(n) F(n + 1) - --- F(n) F(n + 1) + --- F(n) F(n + 1) 638 319 638 29322 4 2 235 4 3 95481 4 6 + ----- F(n) F(n + 1) - --- F(n) F(n + 1) + ----- F(n) F(n + 1) 1595 638 3190 245526 5 28111 8 2 80906 9 - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) + ----- F(n) F(n + 1) 1595 58 1595 63 10 11 9 + --- F(n + 1) - -- 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, 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 ----- \ G(n) = ) F(j) F(n + 4 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 8 4 2 G(n) = 11/3 F(n) F(n + 1) + 1/3 F(n + 1) - 1/3 + 1/4 F(n + 1) 3 5 6 8 + 8/3 F(n) F(n + 1) - 1/12 F(n + 1) - 1/6 F(n + 1) 5 3 179 10 19 4 4 - 23/6 F(n) F(n + 1) + --- F(n) - -- F(n) F(n + 1) 12 12 4 6 5 5 6 2 + 173/2 F(n) F(n + 1) - 200 F(n) F(n + 1) + 3/4 F(n) F(n + 1) 6 4 7 7 3 + 1847/6 F(n) F(n + 1) + 11/6 F(n) F(n + 1) - 955/3 F(n) F(n + 1) 8 2 9 3 7 + 217 F(n) F(n + 1) - 88 F(n) F(n + 1) - 143/6 F(n) F(n + 1) 8 + 5/12 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, 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(n + 5 j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3890786 6 934475 2 8 47052 G(n) = ------- F(n + 1) - ------ F(n) F(n + 1) - ----- F(n) F(n + 1) 319 319 319 1102702 2 4 2 1522524 5 + ------- F(n) F(n + 1) + 1/58 F(n) F(n + 1) - ------- F(n) F(n + 1) 319 319 7997400 3 7 925687 3 3 - ------- F(n) F(n + 1) - ------ F(n) F(n + 1) 319 319 10319325 9 2 3 + -------- F(n) F(n + 1) + 3/638 F(n) F(n + 1) - 5/638 F(n + 1) 319 5144 2 40 3 44003 2 4132800 10 + ---- F(n) - --- F(n) + ----- F(n + 1) - ------- F(n + 1) 319 319 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, 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 ----- \ 2 G(n) = ) F(j) F(2 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 7 2 5 3 G(n) = -315/2 F(n + 1) + 3/2 - 15/2 F(n) F(n + 1) - 5 F(n) F(n + 1) 3 4 2 2 2 - 655/2 F(n) F(n + 1) + 47/2 F(n) F(n + 1) + 7/2 F(n) F(n + 1) 3 6 4 3 + F(n) F(n + 1) + 385 F(n) F(n + 1) - F(n + 1) - 11/2 F(n) 3 2 + 157 F(n + 1) - 135/2 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, 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(2 n + j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 7 2 2 G(n) = -51 F(n) F(n + 1) + 6 + 2015 F(n) F(n + 1) + 144 F(n) F(n + 1) 8 3 4 2 6 - 825 F(n + 1) - 353 F(n) F(n + 1) + 819 F(n + 1) - 40 F(n) F(n + 1) 3 5 - 1715 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, 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(2 n + j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 1867 8 1268 9 2089 8 G(n) = ---- F(n) - 1/11 + 5/22 F(n) - ---- F(n) + ---- F(n) F(n + 1) 66 33 55 4459 8 14 6 2 9809 7 2 - ---- F(n) F(n + 1) - -- F(n) F(n + 1) - ---- F(n) F(n + 1) 66 11 66 2 2 6859 2 3 90703 6 3 + 3/11 F(n) F(n + 1) + ---- F(n) F(n + 1) + ----- F(n) F(n + 1) 110 330 7 3 2 6 - 2/11 F(n) F(n + 1) + 4/11 F(n) F(n + 1) + 3/2 F(n) F(n + 1) 7 2735 2 7 21719 4 - 7/11 F(n) F(n + 1) - ---- F(n) F(n + 1) - ----- F(n) F(n + 1) 33 330 9 3 5 + 1/11 F(n + 1) - 5/11 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, 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(2 n + j) F(2 n + 4 j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 2 3 3 G(n) = -5/12 F(n) F(n + 1) - 1/6 F(n) F(n + 1) + 1/2 F(n) F(n + 1) 4 167425 10 6 + 1/12 F(n + 1) - 1/4 - ------ F(n + 1) + 26341/2 F(n + 1) 12 2 2 36725 2 8 + 3127/4 F(n + 1) + 65/4 F(n) - ----- F(n) F(n + 1) 12 3 3 3 7 - 3021 F(n) F(n + 1) - 27075 F(n) F(n + 1) - 150 F(n) F(n + 1) 5 9 44077 2 4 - 31147/6 F(n) F(n + 1) + 104425/3 F(n) F(n + 1) + ----- 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, 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(2 n + j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 4 G(n) = 679 F(n + 1) - 60 F(n) F(n + 1) + 37/2 F(n) F(n + 1) 2 2 2 3 3 2 + 142 F(n) F(n + 1) + 19/2 F(n) F(n + 1) - 27 F(n) F(n + 1) 3 7 3 5 - 306 F(n) F(n + 1) + 1700 F(n) F(n + 1) - 1375 F(n) F(n + 1) 8 5 2 6 - 1375/2 F(n + 1) + 8 - 9 F(n + 1) - 100 F(n) F(n + 1) - 2 F(n) + 19/2 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, 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 ----- \ 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) 850753 10 1903059 5 15709 G(n) = ------ F(n + 1) + ------- F(n) F(n + 1) - ----- F(n) F(n + 1) 3740 1870 170 487337 10 549387 2 43 5 1866901 6 - ------ F(n) + ------ F(n) - -- F(n + 1) - ------- F(n + 1) 3740 3740 22 3740 13 41 129 3 2 53039 2 8 - -- F(n) + -- F(n + 1) - --- F(n) F(n + 1) + ----- F(n) F(n + 1) 22 22 22 68 230067 2 4 2 3 28415 9 - ------ F(n) F(n + 1) + 2 F(n) F(n + 1) - ----- F(n) F(n + 1) 748 34 108093 9 87 4 23102 2 - ------ F(n) F(n + 1) + -- F(n) F(n + 1) + ----- F(n + 1) 187 22 85 Proof: Both 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(2 n + j) F(4 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 2 6 2 4 G(n) = 1/2 F(n + 1) + 9/2 F(n) + 7/2 F(n) + 7/2 F(n) F(n + 1) 5 4 2 8 - 8 F(n) F(n + 1) + 19 F(n) F(n + 1) + 77 F(n) F(n + 1) 5 4 2 3 3 6 - 213/2 F(n) F(n + 1) + 5 F(n) F(n + 1) + 24 F(n) F(n + 1) 4 7 2 - 44 F(n) F(n + 1) - 2 F(n) F(n + 1) - 12 F(n) F(n + 1) 6 3 3 3 9 9 + 71 F(n) F(n + 1) - 17 F(n) F(n + 1) - 18 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, 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(2 n + j) F(5 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 5 5 383 6 2 2 G(n) = 87 F(n) F(n + 1) + --- F(n) - 21 F(n) F(n + 1) - 84/5 F(n) 10 3 10 7 2 + 1/2 F(n + 1) - 1/2 F(n + 1) - 11/2 F(n) + 8 F(n) F(n + 1) 5 6 9 + 57/5 F(n) F(n + 1) - 21/2 F(n) F(n + 1) - F(n) F(n + 1) 2 5 3 4 6 + 97/2 F(n) F(n + 1) - 125/2 F(n) F(n + 1) + 83/2 F(n) F(n + 1) 6 4 7 3 8 2 - 51/2 F(n) F(n + 1) - 273 F(n) F(n + 1) + 577/2 F(n) F(n + 1) 9 - 102 F(n) F(n + 1) - 27/5 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, 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 ----- \ 2 G(n) = ) F(j) F(2 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 8 4 15 7 G(n) = 7/44 F(n + 1) - 5/11 F(n + 1) - -- F(n) F(n + 1) 11 2421 6 3 35 6 2 2031 8 + ---- F(n) F(n + 1) - -- F(n) F(n + 1) + ---- F(n) F(n + 1) 22 22 44 1037 7 2 339 4 5 105 4 4 - ---- F(n) F(n + 1) + --- F(n) F(n + 1) + --- F(n) F(n + 1) 11 11 44 31 3 5 2 7 36 5 3 - -- F(n) F(n + 1) + 3/4 F(n) F(n + 1) + -- F(n) F(n + 1) 11 11 5 106 9 8 31 25 + 7/44 F(n + 1) - --- F(n) - 3/44 F(n) + -- - -- F(n + 1) 11 44 44 123 3 6 859 5 4 - --- 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, 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(2 n + 2 j) F(2 n + 3 j) / ----- j = 0 then we have the following closed-form expression for G(n) 4 3 3 2 2 G(n) = -1/12 F(n) + 1/6 - 2919/2 F(n) F(n + 1) + 1/3 F(n) F(n + 1) 10 179 2 3 5 - 1/6 F(n + 1) + --- F(n) + 1/3 F(n) F(n + 1) + 10499/6 F(n) F(n + 1) 12 7 5 3 5 5 - 1/3 F(n) F(n + 1) - 2 F(n) F(n + 1) - 51037/6 F(n) F(n + 1) 4 4 36725 4 6 - 442/3 F(n) F(n + 1) - 2/3 F(n) F(n + 1) - ----- F(n) F(n + 1) 12 3 5 3 7 4 2 + 2 F(n) F(n + 1) + 9335 F(n) F(n + 1) + 3111/4 F(n) F(n + 1) 7 9 2 4 + 1/3 F(n) F(n + 1) - 4805/3 F(n) F(n + 1) + 8693/3 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, 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(2 n + 2 j) F(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 13 9 9 53 7 2 15 2 3 G(n) = --- F(n) + 9/22 F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 22 22 22 8979 2 124094 7 3 76421 8 2 - ---- F(n) - ------ F(n) F(n + 1) + ----- F(n) F(n + 1) 550 275 550 31 8 23087 9 6 3 - -- F(n) F(n + 1) - ----- F(n) F(n + 1) + 9/22 F(n) F(n + 1) 22 550 146667 6 4 1201 9 6439 + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) + ---- F(n) F(n + 1) 550 550 275 19331 2 4 25 2 7 47 3 6 + ----- F(n) F(n + 1) + -- F(n) F(n + 1) - -- F(n) F(n + 1) 275 11 22 17 4 5226 5 10 8777 10 - -- F(n) F(n + 1) - ---- F(n) F(n + 1) - 9/22 F(n + 1) + ---- F(n) 22 275 275 Proof: Both 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 ----- \ G(n) = ) F(j) F(2 n + 3 j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 8296 4 2 9 166 2 G(n) = ---- F(n) F(n + 1) - 1/11 F(n + 1) + --- F(n) - 1/11 F(n) 11 11 2 3 133799 5 4 + 3/22 F(n) F(n + 1) - ------ F(n) F(n + 1) + 5/22 F(n) F(n + 1) 22 47 3 2 3 3 8 - -- F(n) F(n + 1) - 1363 F(n) F(n + 1) + 2/11 F(n) F(n + 1) 22 137021 9 3217 338477 2 8 + ------ F(n) F(n + 1) - ---- F(n) F(n + 1) - ------ F(n) F(n + 1) 22 22 22 2 7 57653 2 4 10 + 1/11 F(n) F(n + 1) + ----- F(n) F(n + 1) + 1/11 F(n + 1) 22 41 4 19179 3 7 3 6 + -- F(n) F(n + 1) + ----- F(n) F(n + 1) - 2/11 F(n) F(n + 1) 22 22 137351 4 6 4 5 + ------ F(n) F(n + 1) - 1/11 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, 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 ----- \ 2 G(n) = ) F(j) F(3 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 9 2 6 G(n) = -5825/2 F(n + 1) + 5/2 F(n) - 105/2 F(n + 1) - 17 F(n) F(n + 1) 2 2 7 - 23/2 F(n) + 102 F(n + 1) + 53 F(n + 1) - 625 F(n) F(n + 1) 3 3 8 4 - 130 F(n) F(n + 1) + 14525/2 F(n) F(n + 1) - 2437/2 F(n) F(n + 1) 3 6 3 2 2 4 - 11325/2 F(n) F(n + 1) - 479 F(n) F(n + 1) + 45/2 F(n) F(n + 1) 2 3 5 5 + 736 F(n) F(n + 1) + 120 F(n) F(n + 1) + 2810 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, 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 + j) F(3 n + 2 j) / ----- j = 0 then we have the following closed-form expression for G(n) 10 9 8 2 G(n) = -4 F(n + 1) + 81/2 F(n) F(n + 1) + 525 F(n) F(n + 1) 7 3 5 5 4 6 - 935/2 F(n) F(n + 1) - 371/2 F(n) F(n + 1) + 174 F(n) F(n + 1) 5 5 10 - 235/2 F(n) F(n + 1) + 30 F(n) F(n + 1) + 15 F(n) - 14 F(n) F(n + 1) 2 + 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, 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 + j) F(4 n + j) / ----- j = 0 then we have the following closed-form expression for G(n) 3 3 2 3 7 G(n) = 1/2 F(n + 1) - 9/2 F(n) + 39/2 F(n) + 1174 F(n) F(n + 1) 10 2 4 6 - 1/2 F(n + 1) - 205/2 F(n) F(n + 1) + 24149/2 F(n) F(n + 1) 2 8 3 3 3 4 - 15287 F(n) F(n + 1) - 1466 F(n) F(n + 1) - 125/2 F(n) F(n + 1) 4 2 4 3 2 + 877 F(n) F(n + 1) + 525/2 F(n) F(n + 1) + 59/2 F(n) F(n + 1) 2 4 2 5 + 5211/2 F(n) F(n + 1) - 225 F(n) F(n + 1) - 177 F(n) F(n + 1) 5 6 9 - 5969 F(n) F(n + 1) + 100 F(n) F(n + 1) + 6151 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. ------------------------------------------------ I hope that you enjoyed, dear reader, these, 626, beautiful and deep theorems. 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