Definition:Heptagonal Number

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Definition

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


They can be denoted:

$H_1, H_2, H_3, \ldots$


Definition 1

$H_n = \begin{cases} 0 & : n = 0 \\ H_{n-1} + 5 n - 4 & : n > 0 \end{cases}$


Definition 2

$\displaystyle H_n = \sum_{i \mathop = 1}^n \left({5 i - 4}\right) = 1 + 6 + \cdots + \left({5 \left({n - 1}\right) - 4}\right) + \left({5 n - 4}\right)$


Definition 3

$\forall n \in \N: H_n = P \left({7, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({7, n - 1}\right) + 5 \left({n - 1}\right) + 1 & : n > 0 \end{cases}$

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


Examples of Heptagonal Numbers

The first few heptagonal numbers are as follows:


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Sequence of Heptagonal Numbers

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

$0, 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, \ldots$


Also known as

Heptagonal numbers are also known as heptagon numbers.

Or we can just say that a number is heptagonal.


Some sources denote the $n$th heptagonal number as $\map H n$ in preference to $H_n$.


Also see

  • Results about heptagonal numbers can be found here.


Sources