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A054472 Number of ways to color faces of an icosahedron using at most n colors.

%I A054472 #54 Jul 02 2025 16:01:59
%S A054472 0,1,17824,58130055,18325477888,1589459765875,60935989677984,
%T A054472 1329871177501573,19215358684143616,202627758536996445,
%U A054472 1666666669200004000,11212499922098481787,63895999889747261952,316749396282749868607,1394470923827552301472,5542094550277768379625
%N A054472 Number of ways to color faces of an icosahedron using at most n colors.
%C A054472 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 A054472 From _Robert A. Russell_, Oct 19 2020: (Start)
%C A054472 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 A054472 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 A054472   Conjugacy Class    Count    Even Cycle Indices
%C A054472   Identity              1     x_1^20
%C A054472   Vertex rotation      20     x_1^2x_3^6
%C A054472   Edge rotation        15     x_2^10
%C A054472   Small face rotation  12     x_5^4
%C A054472   Large face rotation  12     x_5^4 (End)
%H A054472 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolyhedronColoring.html">Polyhedron Coloring</a>
%H A054472 <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 A054472 a(n) = (1/60)*(n^20+15*n^10+20*n^8+24*n^4).
%F A054472 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 A054472 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 A054472 a(n) = A252704(n) + A337959(n) = 2*A252704(n) - A337960(n) = 2*A337959(n) + A337960(n). - _Robert A. Russell_, Oct 19 2020
%p A054472 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 A054472 Table[(n^20+15n^10+20n^8+24n^4)/60,{n,0,15}] (* _Harvey P. Dale_, Nov 04 2011 *)
%t A054472 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 A054472 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 A054472 easy,nonn,nice
%O A054472 0,3
%A A054472 _Vladeta Jovovic_, May 20 2000
%E A054472 More terms from _James Sellers_, May 23 2000
%E A054472 More terms from _Colin Barker_, Jul 12 2013