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 sequence is a multiple of the triangular numbers plus 1. For example, the centered square numbers are four times the triangular numbers plus 1.
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
1
5
13
25
Centered hexagonal numbers
1
7
19
37
Formula
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 can be mathematically represented by
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 numberp cannot be a polygonal number (except of course that each p is the second p-agonal number), many centered polygonal numbers are primes.
References
Neil Sloane & Simon Plouffe (1995). The Encyclopedia of Integer Sequences. San Diego: Academic Press.: Fig. M3826
