Centered number
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Centred numbers are 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 any side of 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
- centered triangular numbers 1,4,10,19,31,...
- centered square numbers 1,5,13,25,41,...
- centered pentagonal numbers 1,6,16,31,51,...
- centered hexagonal numbers 1,7,19,37,61,...
- etc...
Each series can be formed by adding 1 to a fixed multiple of the previous triangular number, or to put it algebraically, the nth centered k-gonal number is obtained by the formula
- <math>Ck_n = kT_{n-1}+1<math>
where Tn is the nth triangular number.
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
- <math>{k^3-k^2+2}\over2<math>
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 regular polygonal number (except of course the second k-agonal number), primes occur often enough in the sequences of centered polygonal numbers.
See also
External links
Centered Polygonal Numbers-a large list-clickable. (http://www.virtuescience.com/centered-polygonal.html|)
Centered Polygonal Number -- from MathWorld (http://mathworld.wolfram.com/CenteredPolygonalNumber.html|)
fr:Nombre centré