Generalization of Multi-dimensional Polygonal Numbers

Polygonal Numbers

In the article the formal characterization of triangular numbers (famous from [15] and words “EYPHKA! num = ∆ + ∆ + ∆”) [17] is given. Our primary aim was to formalize one of the items (#42) from Wiedijk’s Top 100 Mathematical Theorems list [33], namely that the sequence of sums of reciprocals of triangular numbers converges to 2. This Mizar representation was written in 2007. As the Mizar lang...

On Universal Sums of Polygonal Numbers

For m = 3, 4, . . . , the polygonal numbers of order m are given by pm(n) = (m−2) ` n 2 ́ +n (n = 0, 1, 2, . . . ). For positive integers a, b, c and i, j, k > 3 with max{i, j, k} > 5, we call the triple (api, bpj , cpk) universal if for any n = 0, 1, 2, . . . there are nonnegative integers x, y, z such that n = api(x)+bpj(y)+cpk(z). We show that there are only 95 candidates for universal triple...

On Wiener numbers of polygonal nets

A Generalization of Stirling Numbers

are called a weighted Stirling pair if f(g(x)) = g(f(x)) = x, i.e.,/and g are reciprocal. When W(x) = l, B^in.k) and B2(n,k) reduce to a Stirling type pair whose properties are exhibited in [7]. In this paper, we shall present a weighted Stirling pair that includes some previous generalizations of Stirling numbers as particular cases. Some related combinatorial and arithmetic properties are als...

Magic Numbers in Polygonal and Polyhedral Clusters

Certain values for the nuclearity of clusters tend to occur more frequently than others. This paper gives a mathematical derivation of the preferred values of nuclearity (the “magic numbers”) corresponding to various symmetrical, close-packed polygons (in two dimensions) and polyhedra (in three dimensions). Simple general formulas are obtained for the numbers of atoms in these clusters. Explici...