Centered polygonal number

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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

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 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

C_{k,n} = 1 + k\cdot T_{n-1} = 1 + kn(n-1)/2.

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

{k^3-k^2+2}\over2

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.

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