Combinatorial proofs of Honsberger-type identities

Abstract

In this article, we consider some generalizations of Fibonacci numbers. We consider k-Fibonacci numbers (that follow the recurrence rule F-k,F- n+2 = kF(k, n+1) + F-k,F- n), the (k, l)-Fibonacci numbers (that follow the recurrence rule F-k,F- n+2 = kF(k, n+1) + F-k,F- n), and the Fibonacci p-step numbers (F-p(n) = F-p(n - 1) + F-p(n - 2)+...+F-p(n-p), with n>p + 1, and p>2). Then we provide combinatorial interpretations of these numbers as square and domino tilings of n-boards, and by easy combinatorial arguments Honsberger identities for these Fibonacci-like numbers are given. While it is a straightforward task to prove these identities with induction, and also by arithmetical manipulations such as rearrangements, the approach used here is quite simple to follow and eventually reduces the proof to a counting problem.