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