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A000545
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Number of ways of n-coloring a dodecahedron.
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8
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1, 96, 9099, 280832, 4073375, 36292320, 230719293, 1145393152, 4707296613, 16666924000, 52307593239, 148602435840, 388302646355, 944900450144, 2162441849625, 4691253854208, 9710376716137, 19280531603808, 36888593841475, 68266682784000, 122597146773927
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OFFSET
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1,2
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COMMENTS
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More explicitly, a(n) is the number of colorings with at most n colors of the faces of a regular dodecahedron, inequivalent under the action of the rotation group of the dodecahedron. It is also the number of inequivalent colorings of the vertices of a regular icosahedron using at most n colors. - José H. Nieto S., Jan 19 2012
Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbols for the regular icosahedron and regular dodecahedron are {3,5} and {5,3} respectively. They are mutually dual. There are 60 elements in the rotation group of the regular dodecahedron/icosahedron. They divide into five conjugacy classes. The first formula is obtained by averaging the dodecahedron face (icosahedron vertex) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
Conjugacy Class Count Even Cycle Indices
Identity 1 x_1^12
Edge rotation 15 x_2^6
Vertex rotation 20 x_3^4
Small face rotation 12 x_1^2x_5^2
Large face rotation 12 x_1^2x_5^2 (End)
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).
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FORMULA
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G.f.: x*((1+x)*(1+x*(82+x*(7847+x*(161900+x*(943640+x*(1764740+x*(943640+x*(161900+x*(7847+x*(82+x)))))))))))/(1-x)^13. - Harvey P. Dale, Apr 25 2011
a(n) = (n^12 + 15*n^6 + 44*n^4) / 60.
a(n) = 1*C(n,1) + 94*C(n,2) + 8814*C(n,3) + 245008*C(n,4) + 2759250*C(n,5) + 15884004*C(n,6) + 52701264*C(n,7) + 106866144*C(n,8) + 134719200*C(n,9) + 103118400*C(n,10) + 43908480*C(n,11) + 7983360*C(n,12), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
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MAPLE
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(1/60)*n^12+(1/4)*n^6+(11/15)*n^4;
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MATHEMATICA
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Table[n^12/60+n^6/4+11 n^4/15, {n, 20}] (* or *) CoefficientList[Series[ -(((1+x) (1+x (82+x (7847+x (161900+x (943640+x (1764740+x (943640+x (161900+x (7847+x (82+x)))))))))))/(x-1)^13), {x, 0, 20}], x] (* Harvey P. Dale, Apr 25 2011 *)
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CROSSREFS
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Other elements: A054472 (dodecahedron vertices, icosahedron faces), A282670 (edges).
Other polyhedra: A006008 (tetrahedron), A047780 (cube faces, octahedron vertices), A000543 (octahedron faces, cube vertices).
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KEYWORD
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nonn,easy
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AUTHOR
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Clint. C. Williams (Clintwill(AT)aol.com)
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STATUS
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approved
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