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A054472 Number of ways to color faces of a icosahedron using at most n colors. 3
0, 1, 17824, 58130055, 18325477888, 1589459765875, 60935989677984, 1329871177501573, 19215358684143616, 202627758536996445, 1666666669200004000, 11212499922098481787, 63895999889747261952, 316749396282749868607, 1394470923827552301472, 5542094550277768379625 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

More explicitly, a(n) is the number of colorings with at most n colors of the faces of a regular icosahedron, inequivalent under the action of the rotation group of the icosahedron. It is also the number of inequivalent colorings of the vertices of a regular dodecahedron using at most n colors. - José H. Nieto S., Jan 19 2012

LINKS

Table of n, a(n) for n=0..15.

Eric Weisstein's World of Mathematics, Polyhedron Coloring

FORMULA

a(n)=(1/60)*(n^20+15*n^10+20*n^8+24*n^4).

G.f.: -x*(x +1)*(x^18 +17802*x^17 +57738159*x^16 +17050750284*x^15 +1199757591558*x^14 +30128721042672*x^13 +329847884196810*x^12 +1749288479932404*x^11 +4727182539811968*x^10 +6598854419308684*x^9 +4727182539811968*x^8 +1749288479932404*x^7 +329847884196810*x^6 +30128721042672*x^5 +1199757591558*x^4 +17050750284*x^3 +57738159*x^2 +17802*x +1) / (x -1)^21. - Colin Barker, Jul 13 2013

a(n)=C(n,1)+17822(n,2)+58076586C(n,3)+18093064608C(n,4)+1498413498750C(n,5)

  +51672950917308C(n,6)+936058547290608C(n,7)+10194866756893728C(n,8)

  +72644237439379200C(n,9)+357895538663241600C(n,10)

  +1264592451488446080C(n,11)+3281293750348373760C(n,12)

  +6337930306906598400C(n,13)+9157388718839961600C(n,14)

  +9858321678965760000C(n,15)+7794071905639219200C(n,16)

  +4394429252269056000C(n,17)+1672620130621440000C(n,18)

  +385209484627968000C(n,19)+40548366802944000C(n,20).  Each term indicates the number of ways to use n colors to color the icosahedron faces (dodecahedron vertices) with exactly 1, 2, ..., 19, or 20 colors.

MAPLE

A054472:=n->(n^20 + 15*n^10 + 20*n^8 + 24*n^4)/60; seq(A054472(n), n=0..15); # Wesley Ivan Hurt, Jan 28 2014

MATHEMATICA

Table[(n^20+15n^10+20n^8+24n^4)/60, {n, 0, 15}] (* Harvey P. Dale, Nov 04 2011 *)

CROSSREFS

Cf. A006008, A047780, A000543, A000545.

Sequence in context: A250333 A203920 A236649 * A204742 A235782 A235787

Adjacent sequences:  A054469 A054470 A054471 * A054473 A054474 A054475

KEYWORD

easy,nonn,nice

AUTHOR

Vladeta Jovovic, May 20 2000

EXTENSIONS

More terms from James A. Sellers, May 23 2000

More terms from Colin Barker, Jul 12 2013

STATUS

approved

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Last modified January 20 03:32 EST 2019. Contains 319323 sequences. (Running on oeis4.)