A Unified Approach to Half-Companion Sequences in Diophantine Triples with Centered (4m+2)-Gonal Numbers

Abstract

In this study, we develop a framework for generating generalized half-companion sequences linked to Diophantine triples that embed centered (4m+2)-gonal numbers across diverse ranks and structural configurations. We further examine the potential extension of these triples into Diophantine quadruples for specific ranks, employing a customized algebraic algorithm to facilitate the analysis. The investigation unveils a spectrum of intriguing numerical phenomena, which are systematically explored and illustrated through MATLAB-driven simulations and Python Programming.