On the Characterization and Enumeration of Some Generalized Trapezoidal Numbers

Abstract

A trapezoidal number, a sum of at least two consecutive positive integers, is a figurate number that can be represented by points rearranged in the plane as a trapezoid. Such numbers have been of interest and extensively studied. In this paper, a generalization of trapezoidal numbers has been introduced. For each positive integer m, a positive integer N is called an m-trapezoidal number if N can be written as an arithmetic series of at least 2 terms with common difference m. Properties of m-trapezoidal numbers have been studied together with their trapezoidal representations. In the special case where m=2, the characterization and enumeration of such numbers have been given as well as illustrative examples. Precisely, for a fixed 2-trapezoidal number N, the ways and the number of ways to write N as an arithmetic series with common difference 2 have been determined. Some remarks on 3-trapezoidal numbers have been provided as well.