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Brainpolice (talk | contribs) Added derivation of relationship 7N(n) + 3 = T(7n-3) (in lieu of a citation, which I could not find.) |
Brainpolice (talk | contribs) Added {{Lead too long}} and {{Citation needed}} |
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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 |title=Figurate Numbers |date=2012 |publisher=World Scientific Publishing Co. |isbn=9814355488 |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 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. |accessdate=3 July 2020}}</ref>:
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==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. {{citation needed}}
==See also==
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