%I #27 Oct 13 2022 14:43:17
%S 1,4,51,4324,2105872,4295327872,35184441295872,1152921514807410688,
%T 151115727460762179076096,79228162514269263405644775424,
%U 166153499473114502703835144588886016,1393796574908163946385532211334573052657664
%N The number of black and white n X n grids distinct under reflections, rotations, and flipping color.
%C 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).
%C Thanks to _Benoit Jubin_ and _Graeme McRae_ for applying Burnside's Lemma appropriately.
%H Isaac E. Lambert, <a href="/A182044/b182044.txt">Table of n, a(n) for n = 1..32</a>
%F a(2n) = (6*2^(2*n^2) + 4*2^(n^2) + 2*2^(n*(2*n+1)) + 2^(4*n^2)) / 16,
%F 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.
%e For n = 2 the a(2) = 4 grids are:
%e ww wb wb ww
%e ww ww bw bb
%p 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)
%p 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)
%p fi;
%p map(f, [$1..16]); # _Robert Israel_, Jul 12 2015
%t 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))];
%t Array[a, 16] (* _Jean-François Alcover_, Apr 10 2019, from Maple *)
%Y Cf. A357536.
%K nonn
%O 1,2
%A _Isaac E. Lambert_, Apr 08 2012
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