Centered polygonal number

In mathematics, the centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers of dots with a constant number of sides.

The following diagrams show a few examples of centered polygonal numbers and their geometric construction.

As can be seen in the above diagrams, the nth centered k-gonal number can be obtained by placing k copies of the (n−1)th triangular number around a central point; therefore, the nth centered k-gonal number is equal to The difference of the n-th and the (n+1)-th consecutive centered k-gonal numbers is k(2n+1).

In fact, if k ≥ 3, k ≠ 8, k ≠ 9, then there are infinitely many centered k-gonal numbers which are primes (assuming the Bunyakovsky conjecture).

The sum of reciprocals for the centered k-gonal numbers is[1]

Proof that centered octa­gonal numbers are odd squares
As the sum of the first n hex numbers is n 3 , the n -th hex number is n 3 − ( n −1) 3