Classes of Gap Balancing Numbers

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 ith 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. Definition 2.1. Let k ≥ 0 be an integer. Define a positive integer B to be an upper k-gap balancing number if 1 + 2 + 3 + • • • + (B − k) = (B + 1) + • • • + (B + r)