Centered polygonal number

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,... OEIS|id=A005448
*centered square numbers 1,5,13,25,41,... (OEIS2C|id=A001844)
*centered pentagonal numbers 1,6,16,31,51,... (OEIS2C|id=A005891)
*centered hexagonal numbers 1,7,19,37,61,... (OEIS2C|id=A003215)
*centered heptagonal numbers 1,8,22,43,71,... (OEIS2C|id=A069099)
*centered octagonal numbers 1,9,25,49,81,... (OEIS2C|id=A016754)
*centered nonagonal numbers 1,10,28,55,91,... (OEIS2C|id=A060544)
*centered decagonal numbers 1,11,31,61,101,... (OEIS2C|id=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;Centered hexagonal numbers

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

: $C_{k,n} = 1 + kcdot T_{n-1} = 1 + kn(n-1)/2.$

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.

References

*Neil Sloane & Simon Plouffe, "The Encyclopedia of Integer Sequences". San Diego: Academic Press (1995): Fig. M3826
*mathworld|urlname=CenteredPolygonalNumber|title=Centered polygonal number

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Centered polygonal number

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