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{{Short description|Type of figurate number}}
{{more citations needed|date=August 2012}}
A '''nonagonal number''', (or an '''enneagonal number'''), is a [[figurate number]] that extends the concept of [[triangular number|triangular]] and [[square number]]s to the [[nonagon]] (a nine-sided polygon).<ref>{{cite book |last1=Deza |first1=Elena|author1-link=Elena Deza |title=Figurate Numbers |date=2012 |publisher=World Scientific Publishing Co. |isbn=978-9814355483 |page=2 |edition=1}}</ref> 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 numbersnumber 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:<ref>{{cite web |title=A001106 |url=https://oeis.org/A001106|website=Online Encyclopedia of Integer Sequences |publisher=OEIS Foundation, Inc. |access-date=3 July 2020}}</ref>
 
:<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. {{OEIS|id=A001106}}.▼
 
▲:[[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==Relationship ''N(n)'' give the ''n''thbetween nonagonal number and ''T(n)'' the ''n''th triangular number,numbers==
 
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>
 
:<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}.</math>.
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}}
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[[Category:Figurate numbers]]
 
 
{{num-stub}}
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