This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A337962 #10 Aug 12 2021 10:27:03
%S A337962 1,68,1659,16464,97935,420708,1443197,4198720,10770597,25016740,
%T A337962 53619335,107545296,204013251,369072900,640912665,1074021632,
%U A337962 1744341865,2755557252,4246675123,6401066960,9457144599,13720858404
%N A337962 Number of achiral colorings of the 12 pentagonal faces of a regular dodecahedron or the 12 vertices of a regular icosahedron using n or fewer colors.
%C A337962 An achiral coloring is identical to its reflection. The Schläfli symbols for the regular icosahedron and regular dodecahedron are {3,5} and {5,3} respectively. They are mutually dual.
%C A337962 There are 60 elements in the automorphism group of the regular dodecahedron/icosahedron that are not in the rotation group. They divide into five conjugacy classes. The first formula is obtained by averaging the dodecahedron face (icosahedron vertex) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
%C A337962 Conjugacy Class Count Odd Cycle Indices
%C A337962 Inversion 1 x_2^6
%C A337962 Edge rotation* 15 x_1^4x_2^4 Asterisk indicates that the
%C A337962 Vertex rotation* 20 x_6^2 operation is followed by an
%C A337962 Small face rotation* 12 x_2^1x_10^1 inversion.
%C A337962 Large face rotation* 12 x_2^1x_10^1
%H A337962 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F A337962 a(n) = n^2 * (15*n^6 + n^4 + 44)/60.
%F A337962 a(n) = 1*C(n,1) + 66*C(n,2) + 1458*C(n,3) + 10232*C(n,4) + 31530*C(n,5) + 47892*C(n,6) + 35280*C(n,7) + 10080*C(n,8), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
%F A337962 a(n) = 2*A252705(n) - A000545(n) = A000545(n) - 2*A337961(n) = A252705(n) - A337961(n).
%F A337962 From _Stefano Spezia_, Oct 04 2020: (Start)
%F A337962 G.f.: x*(1+59*x+1083*x^2+3897*x^3+3087*x^4+1083*x^5+59*x^6+x^7)/(1-x)^9.
%F A337962 a(n) = 9*a(n-1)-36*a(n-2)+84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-8) for n > 8.
%F A337962 (End)
%t A337962 Table[(15n^8+n^6+44n^2)/60,{n,30}]
%Y A337962 Cf. A000545 (oriented), A252705 (unoriented), A337961 (chiral).
%Y A337962 Other elements: A337960 (dodecahedron vertices, icosahedron faces), A337953 (edges).
%Y A337962 Other polyhedra: A006003 (tetrahedron), A337898 (cube faces, octahedron vertices), A337897 (octahedron faces, cube vertices).
%K A337962 nonn,easy
%O A337962 1,2
%A A337962 _Robert A. Russell_, Oct 03 2020