\begin{definition}[Definition:Polygonal Number]
'''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'''.
\end{definition}