\begin{definition}[Definition:Octagonal Number/Definition 2]
Octagonal numbers are defined as:
:$\ds O_n = \sum_{i \mathop = 1}^n \paren {6 i - 5} = 1 + 7 + \cdots + \paren {6 \paren {n - 1} - 5} + \paren {6 n - 5}$
for $n = 1, 2, 3, \ldots$
\end{definition}