An Involution about Fibonacci Numbers

Abstract

In “Proofs That Really Count” A. T. Benjamin and J. J. Quinn provided several identities involving Fibonacci numbers and left some unproven ones. They posed questions to find the combinatorial proofs of these identities. Among these identities, the following one caught our attention because of concerning parity: ∑ k = 0 n ( − 1 ) k f k = 1 + ( − 1 ) n f n − 1 . . Krzykowski found that the coverings in an area with dimension 2 × n can be enumerated by Fibonacci numbers. In this paper, we give an involution throughout the interpretation of Fibonacci numbers given by Krzywkowski to explain the above identity and answer the question provided by Benjamin and Quinn.