polygonalnumber

A polygonal number is a number that can be arranged in a regular polygon with a constant number of dots on each side. The nth s-gonal number, where s is the number of sides and n is a positive integer, counts the dots needed to form a regular s-gon with n dots on each side.

The nth s-gonal number is given by the formula P(s,n) = ((s-2)n^2 - (s-4)n)/2. An equivalent form is P(s,n) = (n/2)[(s-2)n - (s-4)]. The first polygonal number is always 1, and the second is s, corresponding to a polygon with two dots on each side.

Common examples include 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) = (3n^2 - n)/2; and hexagonal numbers (s = 6): P(6,n) = 2n^2 - n.

A key property is the first difference between consecutive polygonal numbers: P(s,n+1) − P(s,n) = (s-2)n + 1. This shows the sequence grows quadratically in n, with linear increments. Polygonal numbers are a subset of figurate numbers and are studied in number theory for their geometric and arithmetic structure.

Historically, polygonal numbers appear in ancient and later mathematical writings, and they feature in the broader study of how integers can be represented as sums of polygonal numbers, as captured in Fermat’s polygonal number theorem.