Please use this identifier to cite or link to this item: http://hdl.handle.net/10553/49162
Title: Combinatorial proofs of Honsberger-type identities
Authors: Plaza, A. 
Falcón, S. 
UNESCO Clasification: 120504 Teoría elemental de los números
Keywords: Combinatorial proof
Generalized Fibonacci numbers
Honsberger identities
Issue Date: 2008
Project: Mtm2005-08441-C02-02. Particiones Triangulares y Algoritmos de Refinamiento 
Journal: International Journal of Mathematical Education in Science and Technology 
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.
URI: http://hdl.handle.net/10553/49162
ISSN: 0020-739X
DOI: 10.1080/00207390801986916
Source: International Journal of Mathematical Education in Science and Technology [ISSN 0020-739X], v. 39 (6), p. 785-792
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