CENTERED POLYGONAL NUMBER

(Redirected from Centered 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.
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,...

★ centered heptagonal numbers 1,8,22,43,71,...

★ centered octagonal numbers 1,9,25,49,81,...

★ centered nonagonal numbers 1,10,28,55,91,...

★ centered decagonal numbers 1,11,31,61,101,...
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
★		★   ★   ★   ★   ★		★   ★   ★   ★   ★   ★   ★   ★   ★   ★   ★   ★   ★		★   ★   ★   ★   ★   ★   ★   ★   ★   ★   ★   ★   ★   ★   ★   ★   ★   ★   ★   ★   ★   ★   ★   ★   ★

;Centered hexagonal numbers

1		7		19		37
★		★ ★ ★ ★ ★ ★ ★		★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★		★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★

As can be seen in the above diagrams, the ''n''th centered ''k''-gonal number can be obtained by placing ''k'' copies of the (''n''−1)th triangular number around a central point; therefore, the ''n''th centered ''k''-gonal number can be mathematically represented by
:$C\_\{k,n\}\; =\; 1\; +\; kcdot\; T\_\{n-1\}\; =\; 1\; +\; kn(n-1)/2$
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
:$\{k^3-k^2+2\}over2$
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.

Contents

External links

External links

★