Definition:Polygonal Number

Definition
A polygonal number is an integer defined recursively as follows:

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

Alternatively, the formula can be given as $$P \left({k, n}\right) = P \left({k, n-1}\right) + n \left({k-2}\right) - \left({k-3}\right)$$.

The name comes from the fact that such numbers can be "arranged" into regular polygonal shapes.

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

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

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

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

See Triangular Number.

Also see the Closed Form for Triangular Numbers.

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

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

See Square Number.

Also see the Odd Number Theorem‎.

Square numbers are of course better known for the fact that $$S_n = n^2$$.

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

$$P \left({5, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({5, n-1}\right) + 3 n - 2 & : n > 0 \end{cases}$$

See Pentagonal Number.

Degenerate Case
When $$k = 2$$, 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.