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]
- .
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 denote the nth nonagonal number, and using the formula for the nth triangular number,
- .
Test for nonagonal numbers
- .
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
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.
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