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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
© 2023 MSP (Mathematical Sciences
Publishers).