\begin{definition}[Definition:Octagonal Number/Definition 1]
Octagonal numbers are defined as:
:$O_n = \begin{cases}
0 & : n = 0 \\
O_{n - 1} + 6 n - 5 & : n > 0 \end{cases}$
for $n = 1, 2, 3, \ldots$
\end{definition}