Nonagonal number

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]

${\displaystyle {\frac {n(7n-5)}{2}}}$.

Nonagonal numbers

The first few nonagonal numbers are:

5500, 5781, 6069, 6364, 6666, 6975, 7291, 7614, 7944, 8281, 8625, 8976, 9334, 9699 (sequence A001106 in the OEIS

).

The parity of nonagonal numbers follows the pattern odd-odd-even-even.

Relationship between nonagonal and triangular numbers

Letting ${\displaystyle N_{n}}$ denote the n^th nonagonal number, and using the formula ${\displaystyle T_{n}={\frac {n(n+1)}{2}}}$ for the n^th triangular number,

${\displaystyle 7N_{n}+3=T_{7n-3}}$.

Test for nonagonal numbers

${\displaystyle {\mathsf {Let}}~x={\frac {{\sqrt {56n+25}}+5}{14}}}$.

If x is an integer, then n is the x-th nonagonal number. If x is not an integer, then n is not nonagonal.

See also

• Centered nonagonal number

References

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