\begin{definition}[Definition:Hexagonal Number/Definition 3]
Hexagonal numbers are defined as the sequence:
:$\forall n \in \N: H_n = P \left({6, n}\right) = \begin{cases}
0 & : n = 0 \\
P \left({6, n - 1}\right) + 4 \left({n - 1}\right) + 1 & : n > 0
\end{cases}$
where $P \left({k, n}\right)$ denotes the $k$-gonal numbers.
\end{definition}