Nonagonal number

An enneagonal number is a figurate number that extends the concept of triangular and square numbers to the enneagon (a nine-sided polygon). However, unlike the triangular and square numbers, the patterns involved in the construction of enneagonal numbers are not rotationally symmetrical. Specifically, the nth enneagonal numbers counts the number of dots in a pattern of n nested enneagons, all sharing a common corner, where the ith enneagon in the pattern has sides made of i dots spaced one unit apart from each other. The enneagonal number for n is given by the formula:

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

The first few enneagonal numbers are:

1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 396, 474, 559, 651, 750, 856, 969, 1089, 1216, 1350, 1491, 1639, 1794, 1956, 2125, 2301, 2484, 2674, 2871, 3075, 3286, 3504, 3729, 3961, 4200, 4446, 4699, 4959, 5226, 5500, 5781, 6069, 6364, 6666, 6975, 7291, 7614, 7944, 8281, 8625, 8976, 9334, 9699. (sequence A001106 in the OEIS)

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

Letting N(n) give the nth enneagonal number and T(n) the nth triangular number,

${7N(n)+3=T(7n-3)}.$

Test for enneagonal numbers

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

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

See also

• Centered enneagonal number