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;
A given positive integer m is s-gonal if there exists an integer n ≥ 1 solving m = P(s,n).
In number theory, sgonal numbers are used in problems about representing integers as polygonal numbers and
Historically, polygonal numbers have roots in ancient mathematics and were developed in the study of figurate