Definition:Octagonal Number

From ProofWiki
Jump to navigation Jump to search

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

$\displaystyle O_n = \sum_{i \mathop = 1}^n \left({6 i - 5}\right) = 1 + 7 + \cdots + \left({6 \left({n - 1}\right) - 5}\right) + \left({6 n - 5}\right)$


Definition 3

$\forall n \in \N: O_n = P \left({8, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({8, n - 1}\right) + 6 \left({n - 1}\right) + 1 & : n > 0 \end{cases}$

where $P \left({k, n}\right)$ denotes the $k$-gonal numbers.


Examples of Octagonal Numbers

The first few octagonal numbers are as follows:


OctagonNumbers.png


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

  • Results about octagonal numbers can be found here.


Sources