cp's OEIS Frontend

A276449 Number of 1-orbits of the cyclic group C_4 for a bi-colored square n X n grid with n squares of one color.

%I A276449 #37 Nov 30 2016 01:18:13
%S A276449 1,0,0,4,6,0,0,120,190,0,0,7140,11480,0,0,635376,1028790,0,0,75287520,
%T A276449 122391522,0,0,11143364232,18161699556,0,0,1978369382080,
%U A276449 3230129794320,0,0,409663695276000,669741609663270,0,0,96930293990660064,158625578809472060
%N A276449 Number of 1-orbits of the cyclic group C_4 for a bi-colored square n X n grid with n squares of one color.
%C A276449 The old name was: Number of ways to choose n points from an n X n grid so that they have 90-degree rotational symmetry.
%C A276449 Consider a square n X n grid with n^2 squares. Each of the n^2 squares comes in two colors.
%C A276449 (E.g., an n X n chessboard with only two black fields, or a binary n X n matrix).
%C A276449 There are N(n) = binomial(n^2,n) = A014062(n) such 2-color grids. We are interested in configurations where n squares are colored in one way, say black, and the remaining ones stay white. Only colored grids modulo rotation around some axis perpendicular to the board through its center are of interest. These rotations represent the cyclic group C_4. Under C_4 operations R(90)^k, k=1..4, there will only be orbits of order 1 (colored grids invariant under R(90)^1, hence any rotation) order 2 (two different grids each not invariant under R(90)^1 but R(90)^2 operation, transforming into each other) and order 4 (four different grids each not invariant under R(90)^k for k=1,2,3, but under R(4)^4, transforming into each other). The orbit structure is denoted by 1^(e(n,1)) 2^(e(n,2)) 4^(e(n,4)) with e(n, 2^j) nonnegative integers for j=0,1,2. One has Sum_{j=0,1,2} 2^j*e(n,2^j) = N(n), and Sum_{j=0,1,2} e(n,2^j) which is the total number of orbits, given in A276454(n).
%C A276449 For example, one of the four 1-orbits of 4 X 4 board. (o) white, (+) black:
%C A276449    + o o +
%C A276449    o o o o
%C A276449    o o o o
%C A276449    + o o +  ,
%C A276449 an example of a 2-orbit,
%C A276449    + o + o   o o o +
%C A276449    o o o o   + o o o
%C A276449    o o o o   o o o +
%C A276449    o + o +   + o o o  ,
%C A276449 an example of a 4-orbit,
%C A276449    + + + +   o o o +   o o o o   + o o o
%C A276449    o o o o   o o o +   o o o o   + o o o
%C A276449    o o o o   o o o +   o o o o   + o o o
%C A276449    o o o o   o o o +   + + + +   + o o o .
%C A276449 The present sequence a(n) gives the number of 1-orbits of such 2-colored boards with n squares of one color under C_4.
%H A276449 Hong-Chang Wang, <a href="/A276449/b276449.txt">Table of n, a(n) for n = 1..100</a>
%F A276449 a(n) = binomial((2*i)^2,i), for n = 4*i,
%F A276449 a(n) = binomial((2*i)*(2*i+1),i), for n = 4*i+1,
%F A276449 a(n) = 0, for others.
%e A276449 a(4) = 4, the arrangements are as follows:
%e A276449    + o o +   o + o o   o o + o   o o o o
%e A276449    o o o o   o o o +   + o o o   o + + o
%e A276449    o o o o   + o o o   o o o +   o + + o
%e A276449    + o o +   o o + o   o + o o   o o o o
%e A276449 a(5) = 6, the arrangements are as follows:
%e A276449    + o o o +   o + o o o   o o + o o
%e A276449    o o o o o   o o o o +   o o o o o
%e A276449    o o + o o   o o + o o   + o + o +
%e A276449    o o o o o   + o o o o   o o o o o
%e A276449    + o o o +   o o o + o   o o + o o
%e A276449    and
%e A276449    o o o + o   o o o o o   o o o o o
%e A276449    + o o o o   o + o + o   o O + o o
%e A276449    O o + o o   o o + o o   o + + + o
%e A276449    o o o o +   o + O + O   o o + o o
%e A276449    o + o o o   o o o o o   o o o o o
%e A276449 reformatted - _Wolfdieter Lang_, Oct 02 2016
%p A276449 seq(op([binomial(2*i*(2*i+1),i),0,0,binomial(4*(i+1)^2,i+1)]),i=0..30); # _Robert Israel_, Sep 05 2016
%t A276449 Table[If[MemberQ[{2, 3}, #], 0, Function[i, Binomial[(2 i) (2 i + #), i]]@ Floor[n/4]] &@ Mod[n, 4], {n, 37}] (* _Michael De Vlieger_, Sep 07 2016 *)
%o A276449 (Python)
%o A276449 import math
%o A276449 def nCr(n,r):
%o A276449     f = math.factorial
%o A276449     return f(n) / f(r) / f(n-r)
%o A276449 # main program
%o A276449 for j in range(101):
%o A276449     if j%4 == 0:
%o A276449         a = nCr((j*j/4),(j/4))
%o A276449     elif j%4 == 1:
%o A276449         a = nCr(((j-1)/2)*((j-1)/2+1),((j-1)/4))
%o A276449     else:
%o A276449         a = 0
%o A276449     print(str(j)+" "+str(a))
%Y A276449 Cf. A014062, A276451, A276452, A276454.
%K A276449 nonn,easy
%O A276449 1,4
%A A276449 _Jason Y.S. Chiu_, _Chiang, Tung-Ying_, _Hsiang-An Wang_, _Hong-Chang Wang_, Sep 02 2016
%E A276449 Edited: New name. Old name as a comment. Text substantially changed. _Wolfdieter Lang_, Oct 02 2016