Centered figurate number that represents a nonagon with a dot in the center
A centered nonagonal number (or centered enneagonal number) is a centeredfigurate number that represents a nonagon with a dot in the center and all other dots surrounding the center dot in successive nonagonal layers. The centered nonagonal number for n layers is given by the formula[1]
Multiplying the (n - 1)th triangular number by 9 and then adding 1 yields the nth centered nonagonal number, but centered nonagonal numbers have an even simpler relation to triangular numbers: every third triangular number (the 1st, 4th, 7th, etc.) is also a centered nonagonal number.[1]
Thus, the first few centered nonagonal numbers are[1]
The list above includes the perfect numbers 28 and 496.
All even perfect numbers are triangular numbers whose index is an odd Mersenne prime.[2] Since every Mersenne prime greater than 3 is congruent to 1 modulo 3, it follows that every even perfect number greater than 6 is a centered nonagonal number.
In 1850, Sir Frederick Pollock conjectured that every natural number is the sum of at most eleven centered nonagonal numbers, which has been neither proven nor disproven.[3]
Congruence Relations
All centered nonagonal numbers are congruent to 1 mod 3.
Therefore the sum of any 3 centered nonagonal numbers and the difference of any two centered nonagonal numbers are divisible by 3.
