Definition:Polygonal Number

Definition

Polygonal numbers are those denumerating a collection of objects which can be arranged in the form of an regular polygon.

A polygonal number is an integer defined recursively as follows:

$\forall k \in \Z_{\ge 2}: \forall n \in Z_{\ge 0}: \map P {k, n} = \begin{cases} 0 & : n = 0 \\ \map P {k, n - 1} + \paren {k - 2} \paren {n - 1} + 1 & : n > 0 \end{cases}$

For a given $k$, polygonal numbers are referred to by the name of the appropriate $k$-sided polygon.

For large $k$, they are therefore called (when used) $k$-gonal numbers.

Also known as

When referring to a $k$-gonal number where $k$ is a more complex expression than just a single number or letter, it may be less unwieldy to refer to it as a polygonal number of order $k$.

Examples

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.

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.

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.

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.

Degenerate Case

Consider the polygonal number $P \left({2, n}\right)$ when $k = 2$.

In this case, the polygon degenerates into a straight line, and the recurrence formula becomes:

$P \left({2, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({2, n-1}\right) + 0 \times \left({n-1}\right) + 1 & : n > 0 \end{cases}$

Hence:

$P \left({2, n}\right) = P \left({2, n-1}\right) + 1$

and the sequence goes:

$0, 1, 2, 3, \ldots$

which is of course the natural numbers.

Historical Note

Polygonal numbers were originally studied by the ancient Greeks.