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 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]

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

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Nonagonal numbers

The first few nonagonal numbers are:

0, 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 nonagonal numbers follows the pattern odd-odd-even-even.

Relationship between nonagonal and triangular numbers

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

7N_{n}+3=T_{7n-3}

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Test for nonagonal numbers

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

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

References

^ Deza, Elena (2012). Figurate Numbers (1 ed.). World Scientific Publishing Co. p. 2. ISBN 978-9814355483.
^ "A001106". Online Encyclopedia of Integer Sequences. OEIS Foundation, Inc. Retrieved 3 July 2020.