A252705 The number of ways to color the faces of a regular dodecahedron with n colors, counting mirror images as one.

1, 82, 5379, 148648, 2085655, 18356514, 116081245, 574795936, 2359033605, 8345970370, 26180606287, 74354990568, 194253329803, 472634761522, 1081541381145, 2346163937920, 4856060529001, 9641643580530, 18446420258299, 34136541925480, 61303301959263

Offset: 1

Author

Robert A. Russell, Dec 20 2014

Keywords

• nonn
• easy

Comments

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.

Examples

			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.
		

References

• 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.

Links

• Colin Barker, 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).

Crossrefs

Cf. A000545 (number when mirror images are counted separately).

Cf. A000332 (tetrahedron), A198833 (cube), A128766 (octahedron), A252704 (icosahedron).

Programs

Mathematica

Table[n^2(n^2+1)(n^8-n^6+16n^4+44)/120,{n,1,30}]

PARI

vector(60, n, n^2*(n^2+1)*(n^8-n^6+16*n^4+44)/120) \\ Michel Marcus, Dec 21 2014

Formula

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) [phone]*C(n,7)[phone]*C(n,8)[phone]*C(n,9)[phone]*C(n,10)[phone]*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.