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The number of inequivalent ways to color the edges of a cube using at most n colors.
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%I #25 Oct 18 2020 09:54:54

%S 1,144,12111,358120,5131650,45528756,288936634,1433251296,5887880415,

%T 20842168600,65402344161,185788177224,485443851256,1181242399260,

%U 2703252560100,5864398969216,12138503871789,24101498435616,46112016365155,85335258695400,153249227870046

%N The number of inequivalent ways to color the edges of a cube using at most n colors.

%C Two edge colorings are equivalent if one is the mirror image of the other or the cube can be picked up and rotated in any manner to obtain the other.

%C The group here has order 48 (compare A060530). - _N. J. A. Sloane_, Aug 14 2012

%C Also the number of unoriented colorings of the 12 edges of a regular octahedron with n or fewer colors. The Schläfli symbols of the cube and octahedron are {4,3} and {3,4} respectively. They are mutually dual. For an unoriented coloring, chiral pairs are counted as one. - _Robert A. Russell_, Oct 17 2020

%H T. D. Noe, <a href="/A199406/b199406.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

%F a(n) = n^12/48 + n^8/16 + n^7/4 + n^6/12 + n^4/6 + n^3/4 + n^2/6.

%F Cycle index = 1/48(s_1^12+3s_1^4s_2^4+12s_1^2s_2^5+4s_2^6+8s_3^4+12s_4^3+8s_6^2).

%F G.f.: -x*(76*x^10 +10016*x^9 +212772*x^8 +1380453*x^7 +3384939*x^6 +3388593*x^5 +1380279*x^4 +211623*x^3 +10317*x^2 +131*x +1)/(x -1)^13. [_Colin Barker_, Aug 13 2012]

%F From _Robert A. Russell_, Oct 17 2020: (Start)

%F a(n) = A060530(n) - A337406(n) = (A060530(n) + A331351(n)) / 2 = A337406(n) + A331351(n).

%F a(n) = 1*C(n,1) + 142*C(n,2) + 11682*C(n,3) + 310536*C(n,4) + 3460725*C(n,5) + 19870590*C(n,6) + 65886660*C(n,7) + 133585200*C(n,8) + 168399000*C(n,9) + 128898000*C(n,10) + 54885600*C(n,11) + 9979200*C(n,12), where the coefficient of C(n,k) is the number of unoriented colorings using exactly k colors. (End)

%t Table[CycleIndex[KSubsetGroup[Automorphisms[CubicalGraph], Edges[CubicalGraph]],s] /. Table[s[i]->n, {i,1,6}], {n,1,15}]

%t Table[(8n^2+12n^3+8n^4+4n^6+12n^7+3n^8+n^12)/48, {n,20}] (* _Robert A. Russell_, Oct 17 2020 *)

%Y Cf. A060530 (oriented), A337406 (chiral), A331351 (achiral), A128766 (cube vertices, octahedron faces), A198833 (cube faces, octahedron vertices), A063842(n-1) (tetrahedron), A337963 (dodecahedron, icosahedron).

%Y Row 3 of A337408 (orthotope edges, orthoplex ridges) and A337412 (orthoplex edges, orthotope ridges).

%K nonn,easy

%O 1,2

%A _Geoffrey Critzer_, Nov 05 2011