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.

Examples

Each sequence is a multiple of the triangular numbers plus 1. For example, the centered square numbers are four times the triangular numbers plus 1.

These series consist of the

• centered triangular numbers 1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, ... (OEIS: A005448)
• centered square numbers 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, ... (OEIS: A001844)
• centered pentagonal numbers 1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, ... (OEIS: A005891)
• centered hexagonal numbers 1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, ... (OEIS: A003215)
• centered heptagonal numbers 1, 8, 22, 43, 71, 106, 148, 197, 253, 316, 386, 463, ... (OEIS: A069099)
• centered octagonal numbers 1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, ... (OEIS: A016754), which are exactly the odd squares
• centered nonagonal numbers 1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, ... (OEIS: A060544), which include all even perfect numbers except 6
• centered decagonal numbers 1, 11, 31, 61, 101, 151, 211, 281, 361, 451, 551, 661, ... (OEIS: A062786)
• centered hendecagonal numbers 1, 12, 34, 67, 111, 166, 232, 309, 397, 496, 606, 727, ... (OEIS: A069125)
• centered dodecagonal numbers 1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, ... (OEIS: A003154), which are exactly the star numbers

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

Formula

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

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

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:

${\frac {k^{2}}{2}}(k-1)+1$

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-gonal number), many centered polygonal numbers are primes. If k is an integer which is ≥ 3 and not equal to 8 or 9, then there are infinitely many centered k-gonal numbers which are primes. (Since all centered octagonal numbers are square numbers, and all centered nonagonal numbers are triangular numbers, thus both of them cannot be prime numbers)

Sum of Reciprocals

The sum of reciprocals for all centered k-gonal numbers is

${\frac {2\pi }{k{\sqrt {1-{\frac {8}{k}}}}}}\tan \left({\frac {\pi }{2}}{\sqrt {1-{\frac {8}{k}}}}\right)$, if k ≠ 8

${\frac {\pi ^{2}}{8}}$, if k = 8

References

• Neil Sloane & Simon Plouffe (1995). The Encyclopedia of Integer Sequences. San Diego: Academic Press.: Fig. M3826
• Weisstein, Eric W. "Centered polygonal number". MathWorld.
• F. Tapson (1999). The Oxford Mathematics Study Dictionary (2nd ed.). Oxford University Press. pp. 88–89. ISBN 0-19-914-567-9.