nonagonal number

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).{{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}} 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:{{cite web |title=A001106 |url=https://oeis.org/A001106|website=Online Encyclopedia of Integer Sequences |publisher=OEIS Foundation, Inc. |access-date=3 July 2020}}

: $\frac {n(7n - 5)}{2}$ .

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 {{OEIS|id=A001106}}.

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 {{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.

References

Category:Figurate numbers

nonagonal number