Definition:Pentagonal Number/Definition 3

Definition

Pentagonal numbers are defined as the sequence:

$\forall n \in \N: P_n = \map P {5, n} = \begin {cases}

0 & : n = 0 \\ \map P {5, n - 1} + 3 \paren {n - 1} + 1 & : n > 0 \end {cases}$ where $\map P {k, n}$ 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 see

• Equivalence of Definitions of Pentagonal Number