Definition:Polygonal Number/Examples

Definition

Triangular Numbers

When $k = 3$, the recurrence relation is:

$\forall n \in \N: T_n = \map P {3, n} = \begin{cases}

0 & : n = 0 \\ \map P {3, n - 1} + \paren {n - 1} + 1 & : n > 0 \end{cases}$ where $\map P {k, n}$ denotes the $k$-gonal numbers.

See Triangular Number.

Square Numbers

When $k = 4$, the recurrence relation is:

$\forall n \in \N: S_n = \map P {4, n} = \begin{cases}

0 & : n = 0 \\ \map P {4, n - 1} + 2 \paren {n - 1} + 1 & : n > 0 \end{cases}$ where $\map P {k, n}$ denotes the $k$-gonal numbers.

See Square Number.

Square numbers are of course better known as:

$S_n = n^2$

Pentagonal Numbers

When $k = 5$, the recurrence relation is:

$\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.

See Pentagonal Number.

Hexagonal Numbers

When $k = 6$, the recurrence relation is:

$\forall n \in \N: H_n = \map P {6, n} = \begin{cases}

0 & : n = 0 \\ \map P {6, n - 1} + 4 \paren {n - 1} + 1 & : n > 0 \end{cases}$ where $\map P {k, n}$ denotes the $k$-gonal numbers.

See Hexagonal Number.

Heptagonal Numbers

When $k = 7$, the recurrence relation is:

$\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.

See Heptagonal Number.

Octagonal Numbers

When $k = 8$, the recurrence relation is:

$\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.

See Octagonal Number.