The Geometry of some Fibonacci Identities in the Hosoya Triangle

TL;DR

This paper explores the geometric structure of the Hosoya triangle, revealing new Fibonacci identities and providing geometric interpretations for classical identities like Cassini and Catalan, extending Pascal triangle properties to Fibonacci products.

Contribution

It introduces geometric interpretations of Fibonacci identities within the Hosoya triangle and extends Pascal triangle identities to Fibonacci product identities.

Findings

Geometric interpretation of Cassini and Catalan identities.

Extension of Pascal triangle properties to Fibonacci products.

New Fibonacci identities derived from geometric analysis.

Abstract

The \emph{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 instance, 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.