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.

These series consist of the

- centered triangular numbers 1,4,10,19,31,... (sequence A005448 in OEIS)
- centered square numbers 1,5,13,25,41,... ( A001844)
- centered pentagonal numbers 1,6,16,31,51,... ( A005891)
- centered hexagonal numbers 1,7,19,37,61,... ( A003215)
- centered heptagonal numbers 1,8,22,43,71,... ( A069099)
- centered octagonal numbers 1,9,25,49,81,... ( A016754)
- centered nonagonal numbers 1,10,28,55,91,... ( A060544)
- centered decagonal numbers 1,11,31,61,101,... ( A062786)

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

1 | 5 | 13 | 25 | |||
---|---|---|---|---|---|---|

- Centered hexagonal numbers

1 | 7 | 19 | 37 | |||
---|---|---|---|---|---|---|

As can be seen in the above diagrams, the *n*th centered *k*-gonal number can be obtained by placing *k* copies of the (*n*−1)th triangular number around a central point; therefore, the *n*th centered *k*-gonal number can be mathematically represented by

- $ C_{k,n} =[\frac{k}{2}](n^2-n)+1. $

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

- $ \frac{k^3-k^2}{2}+1 $

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 (except of course that each *p* is the second *p*-agonal number), many centered polygonal numbers are primes.

## References

- Neil Sloane & Simon Plouffe,
*The Encyclopedia of Integer Sequences*. San Diego: Academic Press (1995): Fig. M3826 - Weisstein, Eric W., "Centered polygonal number" from MathWorld.