Analogues of a Fibonacci-Lucas Identity

Abstract

Sury's 2014 proof of an identity for Fibonacci and Lucas numbers (Identity 236 of Benjamin and Quinn's 2003 book: Proofs that count: The art of combinatorial proof) has excited a lot of comment. We give an alternate, telescoping, proof of this-and associatedidentities and generalize them. We also give analogous identities for other sequences that satisfy a three-term recurrence relation.