A182044 The number of black and white n X n grids distinct under reflections, rotations, and flipping color.

1, 4, 51, 4324, 2105872, 4295327872, 35184441295872, 1152921514807410688, 151115727460762179076096, 79228162514269263405644775424, 166153499473114502703835144588886016, 1393796574908163946385532211334573052657664

Offset

1,2

Comments

a(n) is the number of n X n grids, with each cell painted black or white, distinct under horizontal, vertical, and diagonal reflections, all 3 rotations, and flipping color (changing all white cells to black, and black to white).

Thanks to Benoit Jubin and Graeme McRae for applying Burnside's Lemma appropriately.

Links

Isaac E. Lambert, Table of n, a(n) for n = 1..32

Formula

a(2n) = (6*2^(2*n^2) + 4*2^(n^2) + 2*2^(n*(2*n+1)) + 2^(4*n^2)) / 16,

a(2n+1) = (2^((2*n+1)^2) + 2*2^(1+n*(n+1)) + 2*2^((n+1)*(2*n+1)) + 2^(n*(2*n+2)+1) + 2*2^((2*n+1)*(n+1))) / 16.

Example

For n = 2 the a(2) = 4 grids are:
ww    wb    wb    ww
ww    ww    bw    bb

Maple

f:= n -> if n::even then (3/8)*2^((1/2)*n^2)+(1/4)*2^((1/4)*n^2)+(1/8)*2^((1/2)*n*(n+1))+(1/16)*2^(n^2)
else (1/16)*2^(n^2)+(1/8)*2^(3/4+(1/4)*n^2)+(1/4)*2^((1/2)*n*(n+1))+(1/16)*2^((1/2)*n^2+1/2)
fi;
map(f, [$1..16]); # Robert Israel, Jul 12 2015

Mathematica

a[n_] := If[EvenQ[n], (3*2^(n^2/2))/8 + 2^(n^2/4)/4 + 2^n^2/16 + (1/8)* 2^((1/2)*n*(n+1)), 2^n^2/16 + (1/8)*2^((1/4)*(n^2+3)) + (1/16)*2^((1/2)* (n^2+1)) + (1/4)*2^((1/2)*n*(n+1))];
Array[a, 16] (* Jean-François Alcover, Apr 10 2019, from Maple *)

Crossrefs

Cf. A357536.

Sequence in context: A287231 A289708 A000516 * A000854 A232517 A329009

Adjacent sequences: A182041 A182042 A182043 * A182045 A182046 A182047

Keyword

nonn

Author

Isaac E. Lambert, Apr 08 2012

Status

approved