The 99th Fibonacci identity

Abstract

We provide elementary combinatorial proofs of several Fibonacci and Lucas number identities left open in the book Proofs That Really Count [1], and generalize these to Gibonacci sequences Gn that satisfy the Fibonacci recurrence, but with arbitrary real initial conditions. We offer several new identities as well. Among these, we prove Σk≥0 ( kn)G2k = 5nG2n and Σk≥0 (kn)Gqk(F q-2)n-k = (Fq)nG2n.