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{{Short description|Type of figurate number}}
▲An '''enneagonal''' or '''nonagonal number''' is a [[figurate number]] that extends the concept of [[triangular number|triangular]] and [[square number]]s to the [[nonagon]] (a nine-sided polygon). However, unlike the triangular and square numbers, the patterns involved in the construction of nonagonal numbers are not rotationally symmetrical. Specifically, the ''n''th nonagonal numbers counts the number of dots in a pattern of ''n'' nested nonagons, all sharing a common corner, where the ''i''th 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:
▲:<math>\frac {n(7n - 5)}{2}.</math>
== Nonagonal numbers ==
The first few nonagonal numbers are:
:[[0 (number)|0]], [[1 (number)|1]], [[9 (number)|9]], [[24 (number)|24]], [[46 (number)|46]], [[75 (number)|75]], [[111 (number)|111]], [[154 (number)|154]], [[204 (number)|204]], [[261 (number)|261]], [[325 (number)|325]], [[396 (number)|396]], [[474 (number)|474]], [[559 (number)|559]], [[651 (number)|651]], [[750 (number)|750]], [[856 (number)|856]], [[969 (number)|969]], [[1089 (number)|1089]], [[1216 (number)|1216]], [[1350 (number)|1350]], [[1491 (number)|1491]], [[1639 (number)|1639]], [[1794 (number)|1794]], [[1956 (number)|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
▲:[[1 (number)|1]], [[9 (number)|9]], [[24 (number)|24]], [[46 (number)|46]], [[75 (number)|75]], [[111 (number)|111]], [[154 (number)|154]], [[204 (number)|204]], 261, 325, 396, 474, 559, 651, 750, 856, 969, [[1089 (number)|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. {{OEIS|id=A001106}}
The [[parity (mathematics)|parity]] of nonagonal numbers follows the pattern odd-odd-even-even.
Letting <math>N_n</math> denote the ''n''<sup>th</sup> nonagonal number, and using the formula <math>T_n = \frac{n(n+1)}{2}</math> for the ''n''<sup>th</sup> [[triangular number]],
:<math> 7N_n + 3 = T_{7n-3}</math>.<!-- verify; if you know math you don't need a citation :) -->
==Test for nonagonal numbers==
:<math>\mathsf{Let}~x = \frac{\sqrt{56n+25}+5}{14}
If {{mvar|x}} is an integer, then {{mvar|n}} is the {{mvar|x}}-th nonagonal number. If {{mvar|x}} is not an integer, then {{mvar|n}} is not nonagonal.
==See also==
*[[Centered nonagonal number]]
== References ==
{{reflist}}
{{Figurate numbers}}
{{Classes of natural numbers}}
[[Category:Figurate numbers]]
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