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The geometry of some Fibonacci identities in the Hosoya triangle

Rigoberto Flórez, Robinson A. Higuita and Antara Mukherjee

Vol. 16 (2023), No. 2, 183–200
Abstract

The Hosoya triangle is a triangular array where every entry is a product of two Fibonacci numbers. We use the geometry of this triangle to find new identities related to Fibonacci numbers. We give geometric interpretation for some well-known identities of Fibonacci numbers. For instance, the Cassini identity and the Catalan identity. We also extend some identities that hold in the Pascal triangle to the Hosoya triangle. For example, the hockey stick extends from binomials to products of Fibonacci numbers and the rhombus property extends a binomial identity from the Pascal triangle to an identity of products of Fibonacci numbers in the Hosoya triangle.

Keywords
Fibonacci numbers, Hosoya triangle, rectangle property, zigzag property, braid property, hockey stick property, Cassini identity, Catalan identity
Mathematical Subject Classification 2010
Primary: 11B39
Secondary: 11B83
Milestones
Received: 20 January 2018
Revised: 4 May 2022
Accepted: 7 June 2022
Published: 26 May 2023

Communicated by Kenneth S. Berenhaut
Authors
Rigoberto Flórez
Department of Mathematical Sciences
The Citadel
Charleston, SC
United States
Robinson A. Higuita
Instituto de Matemáticas
Universidad de Antioquia
Medellín
Colombia
Antara Mukherjee
Department of Mathematical Sciences
The Citadel
Charleston, SC
United States