A276454 a(n) = A276452(n) + A276451(n) + A276449(n).

1, 2, 22, 464, 13302, 487152, 21475652, 1106550392, 65221981530, 4327577893800, 319187492622012, 25904823495240144, 2294089575287710984, 220132629099295901408, 22751391952803426496488, 2519687900505935894639088, 297684761086123702744203918

Offset: 1

Author

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

Keywords

• nonn
• easy

Comments

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

The present sequence {a(n)} gives the number of all orbits under C_4 of 2-colored n X n square grids with n squares of one color.

See A054772(n, k) for the table of these total C_4 orbit numbers for 2-colored grids with any number k from {0,1,...,n^2} of squares of one color. - Wolfdieter Lang, Oct 02 2016

Examples

			For n = 4 there are A276449(4) = 4 1-orbits, represented by
   + o o +   o + o o   o o + o   o o o o
   o o o o   o o o +   + o o o   o + + o
   o o o o   + o o o   o o o +   o + + o
   + o o +   o o + o   o + o o   o o o o  .
A276451(4) = 12 2-orbits: one of them is
   + 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 representative.
A276452(4) = 448 4-orbits: one of them is represented by
   + + + +
   o o o o
   o o o o
   o o o o .
The complete orbit structure for n=4 is 1^4 2^12 4^448, see A276449(4) = 4, A276451(4) = 12, A276452(4) = 448.
a(4) = 448 + 12 + 4 = 464.
A014062(4) = 448*4 + 12*2 + 4*1 = 1820.
		

Links

• Hong-Chang Wang, Table of n, a(n) for n = 1..70

Crossrefs

Cf. A014062, A054772, A276451, A276452, A276454.

Programs

Mathematica

f[n_] := If[MemberQ[{2, 3}, #], 0, Function[i, Binomial[(2 i) (2 i + #), i]]@ Floor[n/4]] &@ Mod[n, 4];g[n_] := (Function[j, Binomial[2 j (j + Boole@ OddQ@ n), j]]@ Floor[n/2] - f@ n)/2; Table[(Binomial[n^2, n] - 2 g@ n - f@ n)/4 + (Function[j, Binomial[2 j (j + Boole@ OddQ@ n), j]]@ Floor[n/2] - f@ n)/2 + f@ n, {n, 17}] (* Michael De Vlieger, Sep 12 2016 *)

Python

from math import comb as binomial
for j in range(1, 20):
    t = binomial(j * j, j)
    i = j // 2
    if j % 2 == 0:
        d = binomial(2 * i * i, i)
    else:
        d = binomial(2 * i * (i + 1), i)
    a = (t - d) // 4
    if j % 4 == 0:
        c = binomial((j * j // 4), (j // 4))
    elif j % 4 == 1:
        c = binomial(((j - 1) // 2) * ((j - 1) // 2 + 1), ((j - 1) // 4))
    else:
        c = 0
    b = (d - c) // 2
    print(str(j) + " " + str(a + b + c))

Formula

a(n) = A276452(n) + A276451(n) + A276449(n) for n = 1, 2, 3, ...,

A014062(n) = A276452(n)*4 + A276451(n)*2 + A276449(n).

a(n) = A054772(n, 2), n >= 1. - Wolfdieter Lang, Oct 02 2016

Extensions

Edited by Wolfdieter Lang, Oct 02 2016