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 A199406 The number of inequivalent ways to color the edges of a cube using at most n colors. 10
 1, 144, 12111, 358120, 5131650, 45528756, 288936634, 1433251296, 5887880415, 20842168600, 65402344161, 185788177224, 485443851256, 1181242399260, 2703252560100, 5864398969216, 12138503871789, 24101498435616, 46112016365155, 85335258695400, 153249227870046 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. The group here has order 48 (compare A060530). - N. J. A. Sloane, Aug 14 2012 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 LINKS T. D. Noe, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1). FORMULA a(n) = n^12/48 + n^8/16 + n^7/4 + n^6/12 + n^4/6 + n^3/4 + n^2/6. 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). 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] From Robert A. Russell, Oct 17 2020: (Start) a(n) = A060530(n) - A337406(n) = (A060530(n) + A331351(n)) / 2 = A337406(n) + A331351(n). 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) MATHEMATICA Table[CycleIndex[KSubsetGroup[Automorphisms[CubicalGraph], Edges[CubicalGraph]], s] /. Table[s[i]->n, {i, 1, 6}], {n, 1, 15}] Table[(8n^2+12n^3+8n^4+4n^6+12n^7+3n^8+n^12)/48, {n, 20}] (* Robert A. Russell, Oct 17 2020 *) CROSSREFS Cf. A060530 (oriented), A337406 (chiral), A331351 (achiral), A128766 (cube vertices, octahedron faces), A198833 (cube faces, octahedron vertices), A063842(n-1) (tetrahedron), A337963 (dodecahedron, icosahedron). Row 3 of A337408 (orthotope edges, orthoplex ridges) and A337412 (orthoplex edges, orthotope ridges). Sequence in context: A035821 A231854 A055352 * A182913 A231697 A238932 Adjacent sequences:  A199403 A199404 A199405 * A199407 A199408 A199409 KEYWORD nonn,easy AUTHOR Geoffrey Critzer, Nov 05 2011 STATUS approved

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Last modified June 21 16:39 EDT 2021. Contains 345365 sequences. (Running on oeis4.)