Definition:Polygonal Number

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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.


Examples

Triangular Numbers

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

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

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


See Triangular Number.


Square Numbers

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

$\forall n \in \N: S_n = P \left({4, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({4, n - 1}\right) + 2 \left({n - 1}\right) + 1 & : n > 0 \end{cases}$

where $P \left({k, n}\right)$ 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:

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

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


See Pentagonal Number.


Hexagonal Numbers

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

$\forall n \in \N: H_n = P \left({6, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({6, n - 1}\right) + 4 \left({n - 1}\right) + 1 & : n > 0 \end{cases}$

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


See Hexagonal Number.


Heptagonal Numbers

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

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

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


See Heptagonal Number.


Octagonal Numbers

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

$\forall n \in \N: O_n = P \left({8, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({8, n - 1}\right) + 6 \left({n - 1}\right) + 1 & : n > 0 \end{cases}$

where $P \left({k, n}\right)$ 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.


Also see


Historical Note

Polygonal numbers were originally studied by the ancient Greeks.


Sources