Honeycombs in the Pascal triangle and beyond

TL;DR

This paper introduces a geometric approach to uncovering known and new identities using Pascal and Hosoya triangles, offering fresh insights and techniques for undergraduate research in combinatorics.

Contribution

It presents a novel geometric method to rediscover classical identities and integer sequences in Pascal and Hosoya triangles, expanding traditional combinatorial techniques.

Findings

Identified new identities using geometric patterns

Reinterpreted classical identities through a geometric lens

Provided a new perspective for undergraduate research in combinatorics

Abstract

In this paper we present a geometric approach to discovering some known and some new identities using triangular arrays. Our main aim is to demonstrate how to use the geometric patterns (by Carlitz), in the Pascal and Hosoya triangles to rediscover some classical identities and integer sequences. Therefore, we use new techniques in classical settings which then provide a new perspective in undergraduate research.