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%I #31 Jan 19 2022 21:28:15
%S 2,23,5605504,768614338020786176,
%T 1772303994379887844373479205703254016,
%U 4388012152856549445746584486819723041078276071004502223505850368,746581580725934736852480760189481426040548499078234470565449222456544381939194522144498021170453413888
%N Number of 2-colorings of an n X n X n grid, up to rotational symmetry.
%C The cycle index of the permutation group is given by:
%C Even n: (1/24)*(s_1^n^3 + 8*s_1^n*s_3^((n^3-n)/3) + 6*s_2^(n^3/2) + 6*s_4^(n^3/4) + 3*s_2^(n^3/2));
%C Odd n: (1/24)*(s_1^n^3 + 8*s_1^n*s_3^((n^3-n)/3) + 6*s_1^n*s_2^((n^3-n)/2) + 6*s_1^n*s_4^((n^3-n)/4) + 3*s_1^n*s_2^((n^3-n)/2)).
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cycle_index">Cycle index</a>
%H Paul Oelkers, <a href="/A334616/a334616.jpg">Hand written notes and sketches</a>
%F a(n) = (1/24)*(2^n^3 + 6*2^((n^3)/4) + 9*2^((n^3)/2) + 8*2^((n^3-n)/3+n)) for n even;
%F a(n) = (1/24)*(2^n^3 + 6*2^(((n^3)-n)/4+n) + 9*2^(((n^3)-n)/2+n) + 8*2^(((n^3-n)/3)+n)) for n odd.
%e a(2)=23 from:
%e 00 00
%e 00 00
%e ------------------------------------------
%e 10 00
%e 00 00
%e ------------------------------------------
%e 11 00 10 00 10 01 10 00
%e 00 00 01 00 00 00 00 01
%e ------------------------------------------
%e 11 00 11 00 01 10
%e 10 00 00 10 10 00
%e ------------------------------------------
%e 11 00 11 00 01 10 11 00 11 10
%e 11 00 10 01 10 01 00 11 10 00
%e ------------------------------------------
%e 00 11 00 11 10 01
%e 01 11 11 01 01 11
%e ------------------------------------------
%e 00 11 01 11 01 10 01 11
%e 11 11 10 11 11 11 11 10
%e ------------------------------------------
%e 01 11
%e 11 11
%e ------------------------------------------
%e 11 11
%e 11 11
%e ------------------------------------------
%e An example for the 2-coloring of the 3 X 3 X 3 grid can be written as:
%e 110 000 111
%e 100 000 111
%e 100 000 111
%e This coloring is equivalent to:
%e 111 000 111
%e 001 000 111
%e 000 000 111
%e because you can get this configuration by rotating the first coloring by 90 degrees.
%e But it is different from:
%e 011 000 111
%e 001 000 111
%e 001 000 111
%e because reflections are not considered.
%Y This is the three-dimensional version of A047937.
%Y Cf. A000543.
%K nonn
%O 1,1
%A _Paul Oelkers_, Sep 08 2020
%E More terms from _Stefano Spezia_, Sep 09 2020