Nonagonal number

From Wikipedia, the free encyclopedia
Jump to: navigation, search

A nonagonal number or enneagonal number is a polygonal number that represents a nonagon. The nonagonal number for n is given by the formula:

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

The first few nonagonal 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 OEIS)

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

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

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

Test for nonagonal numbers[edit]

x = \frac{\sqrt{56n+25}+5}{14}.

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