\begin{definition}[Definition:Polygonal 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.
Category:Definitions/Polygonal Numbers
\end{definition}