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,... (OEIS: A005448)
• centered square numbers 1,5,13,25,41,... (OEIS: A001844)
• centered pentagonal numbers 1,6,16,31,51,... (OEIS: A005891)
• centered hexagonal numbers 1,7,19,37,61,... (OEIS: A003215)
• centered heptagonal numbers 1,8,22,43,71,... (OEIS: A069099)
• centered octagonal numbers 1,9,25,49,81,... (OEIS: A016754)
• centered nonagonal numbers 1,10,28,55,91,... (OEIS: A060544, which include all even perfect numbers except 6)
• centered decagonal numbers 1,11,31,61,101,... (OEIS: 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

0		1		5		13

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-agonal number), many centered polygonal numbers are primes.

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.