Nonagonal number: Difference between revisions

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 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>.

:[[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}}.

:<math> 7N_n + 3 = T_{7n-3}.</math>.<!-- verify; if you know math you don't need a citation :) -->

:<math>\mathsf{Let}~x = \frac{\sqrt{56n+25}+5}{14}.</math>.