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A252704 The number of ways to color the faces of a regular icosahedron with n colors, counting mirror images as one. 6
1, 9436, 29131965, 9164844880, 794760482005, 30468267440892, 664937321266057, 9607687940954944, 101313914601247929, 833333459683337020, 5606250353568935653, 31948001059902168528, 158374701054784400173, 697235469002925659548 (list; graph; refs; listen; history; text; internal format)
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.

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

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Last modified July 6 11:37 EDT 2022. Contains 355110 sequences. (Running on oeis4.)