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 A054472 Number of ways to color faces of an 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. - JosÃ© H. Nieto S., Jan 19 2012 LINKS 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*C(n,2) +58076586*C(n,3) +18093064608*C(n,4) +1498413498750*C(n,5) +51672950917308*C(n,6) +936058547290608*C(n,7) +10194866756893728*C(n,8) +72644237439379200*C(n,9) +357895538663241600*C(n,10) +1264592451488446080*C(n,11) +3281293750348373760*C(n,12) +6337930306906598400*C(n,13) +9157388718839961600*C(n,14) +9858321678965760000*C(n,15) +7794071905639219200*C(n,16) +4394429252269056000*C(n,17) +1672620130621440000*C(n,18) +385209484627968000*C(n,19) +40548366802944000*C(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. - Robert A. Russell, Dec 03 2014 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 June 2 12:34 EDT 2020. Contains 334772 sequences. (Running on oeis4.)