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
KEYWORD
nonn
AUTHOR
Isaac E. Lambert, Apr 08 2012
STATUS
approved