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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}}

.

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}

.

Test for nonagonal numbers

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

.

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.

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[1] Deza, Elena (2012). Figurate Numbers (1 ed.). World Scientific Publishing Co. p. 2. ISBN 978-9814355483.

[2] "A001106". Online Encyclopedia of Integer Sequences. OEIS Foundation, Inc. Retrieved 3 July 2020.

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+{{Short description|Type of figurate number}}
-A '''nonagonal number''' or '''enneagonal number''' is a [[polygonal number]] that represents a [[nonagon]].  The nonagonal number for ''n'' is given by the formula:
+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 number counts the 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 {7n^2 - 5n}{2}.</math>
+:<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}}.
+The [[parity (mathematics)|parity]] of nonagonal numbers follows the pattern odd-odd-even-even.
-:[[1 (number)|1]], [[9 (number)|9]], [[24 (number)|24]], [[46 (number)|46]], [[75 (number)|75]], [[111 (number)|111]], [[154 (number)|154]], 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}}
+==Relationship between nonagonal and triangular numbers==
-The [[parity]] of nonagonal numbers follows the pattern odd-odd-even-even.
-Letting ''N(n)'' give the ''n''th nonagonal number and ''T(n)'' the ''n''th triangular number,
+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==
-{{num-stub}}
+:<math>\mathsf{Let}~x = \frac{\sqrt{56n+25}+5}{14}</math>.
-[[Category:figurate numbers]]
+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==
-[[ar:عدد متسع]]
+*[[Centered nonagonal number]]
-[[fr:Nombre ennéagonal]]
-[[fi:Yhdeksänkulmioluku]]
+== References ==
+{{reflist}}
+{{Figurate numbers}}
+{{Classes of natural numbers}}
+[[Category:Figurate numbers]]
+{{num-stub}}