Definition:Polygonal Number

Definition
Polygonal numbers are those denumerating a collection of objects which can be arranged in the form of an regular polygon.

A polygonal number is an integer defined recursively as follows:


 * $\forall k \in \Z_{\ge 2}: \forall n \in Z_{\ge 0}: \map P {k, n} = \begin{cases}

0 & : n = 0 \\ \map P {k, n - 1} + \paren {k - 2} \paren {n - 1} + 1 & : n > 0 \end{cases}$

For a given $k$, polygonal numbers are referred to by the name of the appropriate $k$-sided polygon.

For large $k$, they are therefore called (when used) $k$-gonal numbers.

Also known as
When referring to a $k$-gonal number where $k$ is a more complex expression than just a single number or letter, it may be less unwieldy to refer to it as a polygonal number of order $k$.

Also see

 * Closed Form for Polygonal Numbers