A252704 The number of ways to color the faces of a regular icosahedron with n colors, counting mirror images as one.

1, 9436, 29131965, 9164844880, 794760482005, 30468267440892, 664937321266057, 9607687940954944, 101313914601247929, 833333459683337020, 5606250353568935653, 31948001059902168528, 158374701054784400173, 697235469002925659548

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

Table of n, a(n) for n=1..14.

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

Cf. A054472 (number when mirror images are counted separately).

Cf. A000332 (tetrahedron), A198833 (cube), A128766 (octahedron), A252705 (dodecahedron).

Sequence in context: A068220 A136145 A035792 * A202099 A209854 A235734

Adjacent sequences: A252701 A252702 A252703 * A252705 A252706 A252707

Keyword

nonn,easy

Author

Robert A. Russell, Dec 20 2014

Status

approved