LATTICE POINTS ON THE HOMOGENEOUS CUBIC EQUATION WITH FOUR UNKNOWNS x2 - xy + y2 + 4w2 = 8z3

The Homogeneous cubic equation with four unknowns represented by the equation x2 - xy + y2 + 4w2 = 8z3 is analyzed for its patterns of non zero distinct integral solutions. Here we exhibit four different patterns. In each pattern we can find some interesting relations between the solutions and special numbers like Polygonal number, Three-Dimensional Figurate number, Star number, Rhombic Dodecahedral number etc.

On the Characterization and Enumeration of Some Generalized Trapezoidal Numbers

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.