Definition:Octagonal Number
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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
- Closed Form for Octagonal Numbers: $O_n = n \paren {3 n - 2}$
- Results about octagonal numbers can be found here.
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $8$