A nonagonal number (or an enneagonal number) is a figurate number that extends the concept of triangular and square numbers to the nonagon (a nine-sided polygon).[1] However, unlike the triangular and square numbers, the patterns involved in the construction of nonagonal numbers are not rotationally symmetrical. Specifically, the nth nonagonal number counts the number of dots in a pattern of n nested nonagons, all sharing a common corner, where the ith nonagon in the pattern has sides made of i dots spaced one unit apart from each other. The nonagonal number for n is given by the formula:[2]
The first few nonagonal numbers are:
The parity of nonagonal numbers follows the pattern odd-odd-even-even.
Letting denote the nth nonagonal number, and using the formula for the nth triangular number,
If x is an integer, then n is the x-th nonagonal number. If x is not an integer, then n is not nonagonal.