Definition:Polygonal Number/Examples

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

See Triangular Number.

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:

See Pentagonal Number.