A276451 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.

0, 1, 2, 12, 30, 408, 1012, 17920, 45600, 1059380, 2730756, 78115884, 203235032, 6917206576, 18113945256, 714851008512, 1881039165696, 84449819514060, 223049005408900, 11225502116862880, 29736777118603962, 1658138369930988088, 4403069737450280832

Offset

1,3

Comments

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

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.

Links

Hong-Chang Wang, Table of n, a(n) for n = 1..100

Hong-Chang Wang, Example for n = 4

Formula

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

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

Example

n = 4: one of the two 2-orbits is (o white, + black)
+ o + o   o o o +
o o o o   + o o o
o o o o   o o o +
o + o +   + o o o,
and one can take the first one as a representative.
For n = 3 there are a(3) = 2 2-orbits, represented by
+ o o       o o o
o + o  and  + + +
o o +       o o o.
The orbit structure for n=3 is 1^0 2^2 4^20; see A276449(3) = 0, a(3) = 2, A276452(3) = 20.
For the 12 2-orbits for n=4, see the representatives given in the link.

Mathematica

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 *)

Prog

(Python)
import math
def nCr(n, r):
    f = math.factorial
    return f(n) / f(r) / f(n-r)
# main program
for j in range(101):
    i = j/2
    if j%2==0:
        b = nCr(2*i*i, i)
    else:
        b = nCr(2*i*(i+1), i)
    if j%4==0:
        c = nCr((j*j/4), (j/4))
    elif j%4==1:
        c = nCr(((j-1)/2)*((j-1)/2+1), ((j-1)/4))
    else:
        c = 0
    print(str(j)+" "+str((b-c)/2))

Crossrefs

Cf. A014062, A276449, A276452, A276454.

Sequence in context: A019258 A124903 A197954 * A075352 A116655 A361760

Adjacent sequences: A276448 A276449 A276450 * A276452 A276453 A276454

Keyword

nonn,easy

Author

Chiang, Tung-Ying, Jason Y.S. Chiu, Hong-Chang Wang, Jiangshan Sun, Sep 03 2016

Extensions

Edited: Wolfdieter Lang, Oct 02 2016

Status

approved