------------------------------ Theorem: Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) subject to the initial conditions F(0) = 0, F(1) = 1 Then we have the following identity valid for every non-negative integer n 2 2 2 2 (F(n) - 1 + F(n + 1) F(n) - F(n + 1) ) (F(n) + 1 + F(n + 1) F(n) - F(n + 1) ) = 0 Proof: Routine! (since everything in sight is C-finite). ----------------------------------------------------- Theorem: Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) + F(n - 3) subject to the initial conditions F(0) = 0, F(1) = 0, F(2) = 1 Then we have the following identity valid for every non-negative integer n 3 2 3 2 -1 + F(n + 2) - 2 F(n + 1) F(n + 2) + 2 F(n + 1) - F(n) F(n + 2) 2 2 - 2 F(n) F(n + 1) F(n + 2) + 2 F(n) F(n + 1) + F(n) F(n + 2) 2 3 + 2 F(n) F(n + 1) + F(n) = 0 Proof: Routine! (since everything in sight is C-finite). ----------------------------------------------------- Theorem: Let F(n) be the sequence defined by the recurrence F(n) = F(n - 1) + F(n - 2) + F(n - 3) + F(n - 4) subject to the initial conditions F(0) = 0, F(1) = 0, F(2) = 0, F(3) = 1 Then we have the following identity valid for every non-negative integer n 2 2 1/192 (1 + 2 F(n + 3) F(n) F(n + 2) - 2 F(n + 3) F(n) F(n + 2) 2 2 - F(n + 1) F(n + 2) F(n + 3) - 5 F(n + 3) F(n) F(n + 2) 3 3 3 + 5 F(n + 1) F(n + 2) + F(n + 3) F(n) + 3 F(n + 1) F(n + 2) 3 2 2 3 + 2 F(n + 1) F(n + 3) + 7 F(n + 1) F(n + 2) + 3 F(n + 2) F(n + 3) 3 2 2 3 - 3 F(n + 1) F(n + 3) - 2 F(n + 2) F(n + 3) - 2 F(n + 2) F(n + 3) 2 2 3 3 + 4 F(n) F(n + 1) + 2 F(n) F(n + 1) + 2 F(n) F(n + 2) 3 3 3 2 2 + 3 F(n) F(n + 1) + F(n) F(n + 2) + F(n + 3) F(n) - F(n + 3) F(n) 2 2 - 2 F(n + 3) F(n) F(n + 1) + 7 F(n) F(n + 1) F(n + 2) 4 4 4 - 3 F(n + 3) F(n) F(n + 1) F(n + 2) - F(n + 3) + F(n) - F(n + 1) 4 2 2 + 3 F(n + 2) + 8 F(n) F(n + 1) F(n + 2) + 12 F(n) F(n + 1) F(n + 2) 2 2 - 4 F(n + 3) F(n) F(n + 1) - 3 F(n + 3) F(n) F(n + 1) 2 2 - 2 F(n + 1) F(n + 2) F(n + 3) - 8 F(n + 1) F(n + 2) F(n + 3)) (-1 2 2 + 2 F(n + 3) F(n) F(n + 2) - 2 F(n + 3) F(n) F(n + 2) 2 2 - F(n + 1) F(n + 2) F(n + 3) - 5 F(n + 3) F(n) F(n + 2) 3 3 3 + 5 F(n + 1) F(n + 2) + F(n + 3) F(n) + 3 F(n + 1) F(n + 2) 3 2 2 3 + 2 F(n + 1) F(n + 3) + 7 F(n + 1) F(n + 2) + 3 F(n + 2) F(n + 3) 3 2 2 3 - 3 F(n + 1) F(n + 3) - 2 F(n + 2) F(n + 3) - 2 F(n + 2) F(n + 3) 2 2 3 3 + 4 F(n) F(n + 1) + 2 F(n) F(n + 1) + 2 F(n) F(n + 2) 3 3 3 2 2 + 3 F(n) F(n + 1) + F(n) F(n + 2) + F(n + 3) F(n) - F(n + 3) F(n) 2 2 - 2 F(n + 3) F(n) F(n + 1) + 7 F(n) F(n + 1) F(n + 2) 4 4 4 - 3 F(n + 3) F(n) F(n + 1) F(n + 2) - F(n + 3) + F(n) - F(n + 1) 4 2 2 + 3 F(n + 2) + 8 F(n) F(n + 1) F(n + 2) + 12 F(n) F(n + 1) F(n + 2) 2 2 - 4 F(n + 3) F(n) F(n + 1) - 3 F(n + 3) F(n) F(n + 1) 2 2 - 2 F(n + 1) F(n + 2) F(n + 3) - 8 F(n + 1) F(n + 2) F(n + 3)) = 0 Proof: Routine! (since everything in sight is C-finite). This took, 28.294, seconds .