Definition:Octagonal Number

Definition

Octagonal numbers are those denumerating a collection of objects which can be arranged in the form of a regular octagon.

They can be denoted:

$O_1, O_2, O_3, \ldots$

Definition 1

$O_n = \begin{cases}

0 & : n = 0 \\ O_{n - 1} + 6 n - 5 & : n > 0 \end{cases}$

Definition 2

$\ds O_n = \sum_{i \mathop = 1}^n \paren {6 i - 5} = 1 + 7 + \cdots + \paren {6 \paren {n - 1} - 5} + \paren {6 n - 5}$

Definition 3

$\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 known as

Octagonal numbers are also known as octagon numbers.

Or we can just say that a number is octagonal.

Also see

• Equivalence of Definitions of Octagonal Number

• Closed Form for Octagonal Numbers: $O_n = n \paren {3 n - 2}$

• Definition:Polygonal Number:
  • Definition:Triangular Number
  • Definition:Square Number
  • Definition:Pentagonal Number
  • Definition:Hexagonal Number

• Results about octagonal numbers can be found here.

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $8$