sgonal

sgonal numbers, also called s-gonal numbers, are the polygonal numbers of order s. For any integer s ≥ 3 and n ≥ 1, the nth sgonal number P(s,n) is defined by P(s,n) = ((s−2)n^2 − (s−4)n)/2. This formula yields the familiar sequences: triangular numbers (s=3) P(3,n) = n(n+1)/2; square numbers (s=4) P(4,n) = n^2; pentagonal numbers (s=5) P(5,n) = n(3n−1)/2; hexagonal numbers (s=6) P(6,n) = n(2n−1).

As a function of n, sgonal numbers form a quadratic sequence and grow roughly like (s−2)/2 · n^2; for fixed s they are strictly increasing with n.

A given positive integer m is s-gonal if there exists an integer n ≥ 1 solving m = P(s,n). Equivalently, n must satisfy the quadratic (s−2)n^2 − (s−4)n − 2m = 0, so n = [(s−4) + sqrt((s−4)^2 + 8m(s−2))]/(2(s−2)). Therefore, D = (s−4)^2 + 8m(s−2) must be a perfect square for m to be s-gonal.

In number theory, sgonal numbers are used in problems about representing integers as polygonal numbers and in studying overlaps between different polygonal sequences. They are part of the larger family of figurate numbers, of which polygonal numbers are a subset.

Historically, polygonal numbers have roots in ancient mathematics and were developed in the study of figurate numbers; the term s-gonal reflects the number of sides of the polygon.