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Number of 2-orbits of the cyclic group C_4 for a bi-colored square n X n grid with n squares of one color.
4

%I #47 Nov 10 2016 20:57:32

%S 0,1,2,12,30,408,1012,17920,45600,1059380,2730756,78115884,203235032,

%T 6917206576,18113945256,714851008512,1881039165696,84449819514060,

%U 223049005408900,11225502116862880,29736777118603962,1658138369930988088,4403069737450280832

%N Number of 2-orbits of the cyclic group C_4 for a bi-colored square n X n grid with n squares of one color.

%C For a definition and examples of this problem see the comment section of A276449.

%C The present sequence a(n) gives the number of 2-orbits of such 2-color boards with n squares of one color under C_4.

%H Hong-Chang Wang, <a href="/A276451/b276451.txt">Table of n, a(n) for n = 1..100</a>

%H Hong-Chang Wang, <a href="/A276451/a276451.gif">Example for n = 4</a>

%F a(n) = (binomial(2*i*i,i) - A276449(n))/2, for n = 2*i.

%F a(n) = (binomial(2*i*(i+1),i) - A276449(n))/2, for n = 2*i+1.

%e n = 4: one of the two 2-orbits is (o white, + black)

%e + o + o o o o +

%e o o o o + o o o

%e o o o o o o o +

%e o + o + + o o o,

%e and one can take the first one as a representative.

%e For n = 3 there are a(3) = 2 2-orbits, represented by

%e + o o o o o

%e o + o and + + +

%e o o + o o o.

%e The orbit structure for n=3 is 1^0 2^2 4^20; see A276449(3) = 0, a(3) = 2, A276452(3) = 20.

%e For the 12 2-orbits for n=4, see the representatives given in the link.

%t Table[(Function[j, Binomial[2 j (j + Boole@ OddQ@ n), j]]@ Floor[n/2] - If[MemberQ[{2, 3}, #], 0, Function[i, Binomial[(2 i) (2 i + #), i]]@ Floor[n/4]] &@ Mod[n, 4])/2, {n, 23}] (* _Michael De Vlieger_, Sep 07 2016 *)

%o (Python)

%o import math

%o def nCr(n,r):

%o f = math.factorial

%o return f(n) / f(r) / f(n-r)

%o # main program

%o for j in range(101):

%o i = j/2

%o if j%2==0:

%o b = nCr(2*i*i,i)

%o else:

%o b = nCr(2*i*(i+1),i)

%o if j%4==0:

%o c = nCr((j*j/4),(j/4))

%o elif j%4==1:

%o c = nCr(((j-1)/2)*((j-1)/2+1),((j-1)/4))

%o else:

%o c = 0

%o print(str(j)+" "+str((b-c)/2))

%Y Cf. A014062, A276449, A276452, A276454.

%K nonn,easy

%O 1,3

%A _Chiang, Tung-Ying_, _Jason Y.S. Chiu_, _Hong-Chang Wang_, _Jiangshan Sun_, Sep 03 2016

%E Edited: _Wolfdieter Lang_, Oct 02 2016