centered number

Centred numbers are 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 any side of 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,...
etc...

Each series can be formed by adding 1 to a fixed multiple of the previous triangular number, or to put it algebraically, the nth centered k-gonal number is obtained by the formula

Ck_n = kT_{n-1}+1

where T is a triangular number. 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 regular polygonal number (except of course the second k-agonal number), primes occur often enough in the sequences of centered polygonal numbers.

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