## The 99th Fibonacci identity

### Summary

Summary: 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 n G n G k$\geq 0$ k 2k = 5nG2n and k$\geq 0$ k qk (Fq - 2)n - k = (Fq )nG2n.

05A19, 11B39