\begin{definition}[Definition:Pentagonal Number/Definition 3]
Pentagonal numbers are defined as the sequence:
:$\forall n \in \N: P_n = \map P {5, n} = \begin {cases}
0 & : n = 0 \\
\map P {5, n - 1} + 3 \paren {n - 1} + 1 & : n > 0
\end {cases}$
where $\map P {k, n}$ denotes the $k$-gonal numbers.
\end{definition}