Recurrence sequences in the hyperbolic Pascal triangle corresponding to the regular mosaic $\{4,5\}

TL;DR

This paper explores the structure of recurrence sequences within a hyperbolic Pascal triangle linked to the regular mosaic {4,5}, revealing how binary recursive sequences are represented in this geometric setting.

Contribution

It introduces a novel analysis of recurrence sequences in hyperbolic Pascal triangles associated with the {4,5} mosaic, expanding understanding of their combinatorial and geometric properties.

Findings

Recurrence sequences are represented in the hyperbolic Pascal triangle.

The structure of these sequences relates to the regular mosaic {4,5}.

New connections between hyperbolic geometry and recursive sequences are established.

Abstract

Recently, a new generalization of Pascal's triangle, the so-called hyperbolic Pascal triangles were introduced. The mathematical background goes back to the regular mosaics in the hyperbolic plane. In this article, we investigate the paths in the hyperbolic Pascal triangle corresponding to the regular mosaic $\{4,5\}$, in which the binary recursive sequences $f_{n}=\alpha f_{n-1}\pm f_{n-2}$ are represented ($\alpha\in\mathbb{N}^+$).