Centered polygonal number


The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers with a constant number of sides. Each side of a polygonal layer contains one dot more than a side in the previous layer, so starting from the second polygonal layer each layer of a centered k-gonal number contains k more points than the previous layer.

Examples

Each element in the sequence is a multiple of the previous triangular number plus 1. This can be formalized by the equation where a is the number of sides of the polygon, and x is the sequence number, starting with zero for the initial 1. For example, the centered square numbers are four times the triangular numbers plus 1, or equivalently.
These series consist of the
and so on.
The following diagrams show a few examples of centered polygonal numbers and their geometric construction. Compare these diagrams with the diagrams in Polygonal number.

Centered square numbers

Centered hexagonal numbers

Formula

As can be seen in the above diagrams, the nth centered k-gonal number can be obtained by placing k copies of the th triangular number around a central point; therefore, the nth centered k-gonal number can be mathematically represented by
The difference of the n-th and the -th consecutive centered k-gonal numbers is k.
Just as is the case with regular polygonal numbers, the first centered k-gonal number is 1. Thus, for any k, 1 is both k-gonal and centered k-gonal. The next number to be both k-gonal and centered k-gonal can be found using the formula:
which tells us that 10 is both triangular and centered triangular, 25 is both square and centered square, etc.
Whereas a prime number p cannot be a polygonal number, many centered polygonal numbers are primes. In fact, if k ≥ 3, k ≠ 8, k ≠ 9, then there are infinitely many centered k-gonal numbers which are primes.

Sum of Reciprocals

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