Definition:Polygonal Number/Degenerate Case
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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.