OFFSET
1,2
COMMENTS
The cycle index using the full automorphism group for faces of an icosahedron is (x1^20+15*x2^10+20*x1^2*x3^6+24*x5^4+15*x1^4*x2^8+x2^10+20*x2*x6^3+24*x10^2)/120.
Also the number of ways to color the vertices of a regular dodecahedron with n colors, counting mirror images as one.
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
Index entries for linear recurrences with constant coefficients, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).
FORMULA
a(n) = n^2*(n^18+15*n^10+16*n^8+20*n^6+44*n^2+24)/120.
G.f.: x*(x+1)*(x^18+9414*x^17+28924605*x^16+8526129240*x^15+599877779040*x^14 +15064347905208*x^13+164923977484392*x^12+874644240573864*x^11 +2363591146376826*x^10+3299427410370820*x^9+2363591146376826*x^8 +874644240573864*x^7+164923977484392*x^6+15064347905208*x^5 +599877779040*x^4+8526129240*x^3+28924605*x^2+9414*x+1)/(1-x)^21.
a(n) = C(n,1)+9434*C(n,2)+29103660*C(n,3)+9048373632*C(n,4)+749227482900*C(n,5) +25836594724296*C(n,6)+468029669151744*C(n,7)+5097434180194944*C(n,8) +36322119730219680*C(n,9)+178947770105039040*C(n,10)+632296226073536640*C(n,11)+1640646875234062080*C(n,12)+3168965153453299200*C(n,13)+4578694359419980800*C(n,14)+4929160839482880000*C(n,15)+3897035952819609600*C(n,16) +2197214626134528000*C(n,17)+836310065310720000*C(n,18)+192604742313984000*C(n,19)+20274183401472000*C(n,20). Each term indicates the number of ways to use n colors to color the icosahedron with exactly 1, 2, 3, ..., 18, 19, or 20 colors.
EXAMPLE
For n=2, a(2)=9436, the number of ways to color the faces of a regular icosahedron with two colors, counting mirror images as the same. Of these, two use the same color for all faces, and 9434 use both colors.
MATHEMATICA
Table[n^2(n^18+15n^10+16n^8+20n^6+44n^2+24)/120, {n, 1, 30}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, Dec 20 2014
STATUS
approved