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A252705
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The number of ways to color the faces of a regular dodecahedron with n colors, counting mirror images as one.
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7
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1, 82, 5379, 148648, 2085655, 18356514, 116081245, 574795936, 2359033605, 8345970370, 26180606287, 74354990568, 194253329803, 472634761522, 1081541381145, 2346163937920, 4856060529001, 9641643580530, 18446420258299, 34136541925480, 61303301959263
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OFFSET
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1,2
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COMMENTS
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The cycle index using the full automorphism group for faces of a dodecahedron is (x1^12+15*x2^6+20*x3^4+24*x1^2*x5^2+15*x1^4*x2^4+x2^6+20*x6^2+24*x2*x10)/120.
Also the number of ways to color the vertices of a regular icosahedron with n colors, counting mirror images as one.
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REFERENCES
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F. S. Roberts and B. Tesman, Applied Combinatorics, 2d Ed., Pearson Prentice Hall, 2005, pages 439-488.
J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge University Press, 1992, pages 461-474.
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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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a(n) = n^2*(n^2+1)*(n^8-n^6+16*n^4+44)/120.
G.f.: x*(x+1)*(x^10+68*x^9+4323*x^8+80508*x^7+469548*x^6+886944*x^5+469548*x^4 +80508*x^3+4323*x^2+68*x+1)/(1-x)^13.
a(n) = C(n,1)+80*C(n,2)+5136*C(n,3)+127620*C(n,4)+1395390*C(n,5)+7965948*C(n,6) +26368272*C(n,7)+53438112*C(n,8)+67359600*C(n,9)+51559200*C(n,10)+21954240*C(n,11)+3991680*C(n,12). Each term indicates the number of ways to use n colors to color the dodecahedron with exactly 1, 2, 3, ..., 10, 11, or 12 colors.
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EXAMPLE
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For n=2, a(2)=82, the number of ways to color the faces of a regular dodecahedron with two colors, counting mirror images as the same. Of these, two use the same color for all faces, and 80 use both colors.
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MATHEMATICA
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Table[n^2(n^2+1)(n^8-n^6+16n^4+44)/120, {n, 1, 30}]
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PROG
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(PARI) vector(60, n, n^2*(n^2+1)*(n^8-n^6+16*n^4+44)/120) \\ Michel Marcus, Dec 21 2014
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CROSSREFS
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Cf. A000545 (number when mirror images are counted separately).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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