On the characterization and enumeration of some generalized trapezoidal numbers (Q1751516)

Summary: 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 \(m\)-\textit{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.