Hyperbolic Pascal triangles

TL;DR

This paper introduces a new hyperbolic Pascal triangle derived from hyperbolic plane mosaics, analyzing its properties, patterns, and recurrences, extending classical Pascal's triangle into hyperbolic geometry.

Contribution

It presents the first systematic construction of hyperbolic Pascal triangles from mosaics and explores their combinatorial and algebraic properties.

Findings

Number of elements in each row is determined.

Sum and alternating sum of row elements are analyzed.

Binary recurrences appear in the hyperbolic Pascal triangles.

Abstract

In this paper, we introduce a new generalization of Pascal's triangle. The new object is called the hyperbolic Pascal triangle since the mathematical background goes back to regular mosaics on the hyperbolic plane. We describe precisely the procedure of how to obtain a given type of hyperbolic Pascal triangle from a mosaic. Then we study certain quantitative properties such as the number, the sum, and the alternating sum of the elements of a row. Moreover, the pattern of the rows, and the appearence of some binary recurrences in a fixed hyperbolic triangle are investigated.