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A334616 Number of 2-colorings of an n X n X n grid, up to rotational symmetry.

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