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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

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
RedDot RedDotBlank300RedDot
Blank300GrayDotBlank300
RedDotBlank300RedDot
RedDotBlank300RedDotBlank300RedDot
Blank300GrayDotBlank300GrayDotBlank300
RedDotBlank300GrayDotBlank300RedDot
Blank300GrayDotBlank300GrayDotBlank300
RedDotBlank300RedDotBlank300RedDot
RedDotBlank300RedDotBlank300RedDotBlank300RedDot
Blank300GrayDotBlank300GrayDotBlank300GrayDotBlank300
RedDotBlank300GrayDotBlank300GrayDotBlank300RedDot
Blank300GrayDotBlank300GrayDotBlank300GrayDotBlank300
RedDotBlank300GrayDotBlank300GrayDotBlank300RedDot
Blank300GrayDotBlank300GrayDotBlank300GrayDotBlank300
RedDotBlank300RedDotBlank300RedDotBlank300RedDot
Centered hexagonal numbers
1 7 19 37
RedDotX RedDotRedDotX
RedDotXGrayDotXRedDotX
RedDotXRedDotX
RedDotRedDotXRedDotX
RedDotGrayDotXGrayDotXRedDotX
RedDotGrayDotXGrayDotXGrayDotXRedDotX
RedDotGrayDotXGrayDotXRedDotX
RedDotRedDotXRedDotX
RedDotRedDotRedDotXRedDotX
RedDotGrayDotXGrayDotXGrayDotXRedDotX
RedDotGrayDotXGrayDotXGrayDotXGrayDotXRedDotX
RedDotGrayDotXGrayDotXGrayDotXGrayDotXGrayDotXRedDotX
RedDotGrayDotXGrayDotXGrayDotXGrayDotXRedDotX
RedDotGrayDotXGrayDotXGrayDotXRedDotX
RedDotRedDotRedDotXRedDotX

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{k}{2}](n^2-n)+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^3-k^2}{2}+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


ar:عدد ممركز مضلعit:Numero poligonale centrato

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