# Definition:Pentagonal Number

## Definition

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

They can be denoted:

$P_1, P_2, P_3, \ldots$

### Definition 1

$P_n = \begin{cases} 0 & : n = 0 \\ P_{n-1} + 3 n - 2 & : n > 0 \end{cases}$

### Definition 2

$\displaystyle P_n = \sum_{i \mathop = 1}^n \left({3 i - 2}\right) = 1 + 4 + \cdots + \left({3 \left({n - 1}\right) - 2}\right) + \left({3 n - 2}\right)$

### Definition 3

$\forall n \in \N: P_n = P \left({5, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({5, n - 1}\right) + 3 \left({n - 1}\right) + 1 & : n > 0 \end{cases}$

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

## Examples of Pentagonal Numbers

The first few pentagonal numbers are as follows:

### Sequence of Pentagonal Numbers

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

$0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, \ldots$

## Also known as

Pentagonal numbers are also known as pentagon numbers.

Or we can just say that a number is pentagonal.

## Also see

• Results about pentagonal numbers can be found here.