Some Algebraic Relations on Integer Sequences Involving Oblong and Balancing Numbers

Abstract

Let \(k \geq 0\) be an integer. Oblong (pronic) numbers are numbers of the form \(O_k = k(k+1)\). In this work, we set a new integer sequence \(B = B_n(k)\) defined as \(B_0 = 0\), \(B_1 = 1\), and \(B_n = O_k B_{n-1} – B_{n-2}\) for \(n \geq 2\), and then derive some algebraic relations on it. Later, we give some new results on balancing numbers via oblong numbers.