# Definition:Heptagonal Number

## 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

$\ds H_n = \sum_{i \mathop = 1}^n \paren {5 i - 4} = 1 + 6 + \cdots + \paren {5 \paren {n - 1} - 4} + \paren {5 n - 4}$

### Definition 3

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

where $\map P {k, n}$ denotes the $k$-gonal numbers.

## Examples of Heptagonal Numbers

The first few heptagonal numbers are as follows:

File:HeptagonNumbers.png

### 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.