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.
The series
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 the OEIS)
- 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
| 1 | 5 | 13 | 25 | |||
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Centered hexagonal numbers
| 1 | 7 | 19 | 37 | |||
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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.
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. p. 88-89. ISBN 0-19-914-567-9.