Definition:Octagonal Number/Definition 3

Definition

Octagonal numbers are defined as the sequence:

$\forall n \in \N: O_n = \map P {8, n} = \begin{cases}

0 & : n = 0 \\ \map P {8, n - 1} + 6 \paren {n - 1} + 1 & : n > 0 \end{cases}$ where $\map P {k, n}$ denotes the $k$-gonal numbers.

Examples of Octagonal Numbers

The first few octagonal numbers are as follows:

Sequence of Octagonal Numbers

The sequence of octagonal numbers, for $n \in \Z_{\ge 0}$, begins:

$0, 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, \ldots$

Also see

• Equivalence of Definitions of Octagonal Number