Classes of Gap Balancing Numbers

TL;DR

This paper surveys and extends the study of gap balancing numbers, a generalization of balancing numbers related to triangular numbers, exploring their structure, unifying terminology, and proposing a new conjecture.

Contribution

It unifies existing terminology, extends previous results, and introduces new findings and a conjecture on the classes of gap balancing numbers based on gap size.

Findings

Extended the understanding of gap balancing numbers

Proposed a conjecture on the number of classes based on gap size

Unified terminology and surveyed early results

Abstract

Gap balancing numbers are a certain generalization of balancing and cobalancing numbers that arise from studying the equation ${T(L)+T(B)=T(m)}$ where $T(i)$ is the $i$th triangular number. In this paper, we survey early results, attempt to unify existing terminology, and extend prior findings. In addition, we further investigate the structure of classes of gap balancing numbers and related sequences, presenting new results and a conjecture regarding the number of classes based on the gap size.