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

centered triangular numbers 1,4,10,19,31,...
centered square numbers 1,5,13,25,41,...
centered pentagonal numbers 1,6,16,31,51,...
centered hexagonal numbers 1,7,19,37,61,...
centered heptagonal numbers 1,8,22,43,71,...
centered octagonal numbers 1,9,25,49,81,...
centered nonagonal numbers 1,10,28,55,91,...
centered decagonal numbers 1,11,31,61,101,...

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.