Definition:Polygonal Number/Degenerate Case

Definition
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.