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Number of ways to color faces of an icosahedron using at most n colors.
8

%I #52 Aug 11 2021 19:24:46

%S 0,1,17824,58130055,18325477888,1589459765875,60935989677984,

%T 1329871177501573,19215358684143616,202627758536996445,

%U 1666666669200004000,11212499922098481787,63895999889747261952,316749396282749868607,1394470923827552301472,5542094550277768379625

%N Number of ways to color faces of an icosahedron using at most n colors.

%C 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

%C From _Robert A. Russell_, Oct 19 2020: (Start)

%C Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbols for the regular icosahedron and regular dodecahedron are {3,5} and {5,3} respectively. They are mutually dual.

%C There are 60 elements in the rotation group of the regular dodecahedron/icosahedron. They divide into five conjugacy classes. The first formula is obtained by averaging the icosahedron face (dodecahedron vertex) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.

%C Conjugacy Class Count Even Cycle Indices

%C Identity 1 x_1^20

%C Vertex rotation 20 x_1^2x_3^6

%C Edge rotation 15 x_2^10

%C Small face rotation 12 x_5^4

%C Large face rotation 12 x_5^4 (End)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PolyhedronColoring.html">Polyhedron Coloring</a>

%H <a href="/index/Rec#order_21">Index entries for linear recurrences with constant coefficients</a>, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).

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

%F 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

%F a(n) = 1*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), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors. - _Robert A. Russell_, Dec 03 2014

%F a(n) = A252704(n) + A337959(n) = 2*A252704(n) - A337960(n) = 2*A337959(n) + A337960(n). - _Robert A. Russell_, Oct 19 2020

%p 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

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

%t LinearRecurrence[{21,-210,1330,-5985,20349,-54264,116280,-203490,293930,-352716,352716,-293930,203490,-116280,54264,-20349,5985,-1330,210,-21,1},{0,1,17824,58130055,18325477888,1589459765875,60935989677984,1329871177501573,19215358684143616,202627758536996445,1666666669200004000,11212499922098481787,63895999889747261952,316749396282749868607,1394470923827552301472,5542094550277768379625,20148763660520129167360,67737190111299199134361,212470603607497593076128,626499557627304397693519,1747626666669235200064000},20] (* _Harvey P. Dale_, Aug 11 2021 *)

%Y Cf. A252704 (unoriented), A337959 (chiral), A337960 (achiral), A282670 (edges), A000545 (dodecahedron faces, icosahedron vertices), A006008 (tetrahedron), A047780 (cube faces, octahedron vertices), A000543 (octahedron faces, cube vertices).

%K easy,nonn,nice

%O 0,3

%A _Vladeta Jovovic_, May 20 2000

%E More terms from _James A. Sellers_, May 23 2000

%E More terms from _Colin Barker_, Jul 12 2013