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A276454 a(n) = A276452(n) + A276451(n) + A276449(n). 4
1, 2, 22, 464, 13302, 487152, 21475652, 1106550392, 65221981530, 4327577893800, 319187492622012, 25904823495240144, 2294089575287710984, 220132629099295901408, 22751391952803426496488, 2519687900505935894639088, 297684761086123702744203918 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

LINKS

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

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

EXAMPLE

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 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.

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 *)

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):

   t = nCr(j*j, j)

    i = j/2

    if j%2==0:

        d = nCr(2*i*i, i)

    else:

        d = nCr(2*i*(i+1), i)

    a = (t-d)/4

    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

    b = (d-c)/2

    print(str(j)+" "+str(a+b+c))

CROSSREFS

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

Sequence in context: A328158 A266522 A084949 * A137076 A090730 A090313

Adjacent sequences:  A276451 A276452 A276453 * A276455 A276456 A276457

KEYWORD

nonn,easy

AUTHOR

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

EXTENSIONS

Edited: Wolfdieter Lang, Oct 02 2016

STATUS

approved

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Last modified May 12 22:49 EDT 2021. Contains 343829 sequences. (Running on oeis4.)