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International Journal of Mathematics and Mathematical Sciences
Volume 2017 (2017), Article ID 4515249, 6 pages
https://doi.org/10.1155/2017/4515249
Research Article

On the Characterization and Enumeration of Some Generalized Trapezoidal Numbers

Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom 73000, Thailand

Correspondence should be addressed to Somphong Jitman; moc.liamg@namtijs

Received 4 May 2017; Accepted 16 July 2017; Published 22 August 2017

Academic Editor: Susana Montes

Copyright © 2017 Somphong Jitman and Chakrit Phongthai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 , a positive integer is called an -trapezoidal number if can be written as an arithmetic series of at least terms with common difference . Properties of -trapezoidal numbers have been studied together with their trapezoidal representations. In the special case where , the characterization and enumeration of such numbers have been given as well as illustrative examples. Precisely, for a fixed -trapezoidal number , the ways and the number of ways to write as an arithmetic series with common difference have been determined. Some remarks on -trapezoidal numbers have been provided as well.