cp's OEIS Frontend

A337897 Number of achiral colorings of the 8 triangular faces of a regular octahedron or the 8 vertices of a cube using n or fewer colors.

%I A337897 #19 Mar 08 2024 11:56:21
%S A337897 1,21,201,1076,4025,11901,29841,66256,134001,251725,445401,750036,
%T A337897 1211561,1888901,2856225,4205376,6048481,8520741,11783401,16026900,
%U A337897 21474201,28384301,37055921,47831376,61100625,77305501,96944121
%N A337897 Number of achiral colorings of the 8 triangular faces of a regular octahedron or the 8 vertices of a cube using n or fewer colors.
%C A337897 An achiral coloring is identical to its reflection. The Schläfli symbols for the cube and regular octahedron are {4,3} and {3,4} respectively. They are mutually dual.
%C A337897 There are 24 elements in the automorphism group of the regular octahedron/cube that are not in the rotation group. They divide into five conjugacy classes. The first formula is obtained by averaging the cube vertex (octahedron face) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
%C A337897   Conjugacy Class     Count    Odd Cycle Indices
%C A337897   Inversion              1     x_2^4
%C A337897   Vertex rotation*       8     x_2^1x_6^1      Asterisk indicates that the
%C A337897   Edge rotation*         6     x_1^4x_2^2      operation is followed by an
%C A337897   Small face rotation*   3     x_4^2           inversion.
%C A337897   Large face rotation*   6     x_2^4
%H A337897 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7, -21, 35, -35, 21, -7, 1).
%F A337897 a(n) = n^2 * (7 + 2*n^2 + 3*n^4) / 12.
%F A337897 a(n) = 1*C(n,1) + 19*C(n,2) + 141*C(n,3) + 394*C(n,4) + 450*C(n,5) + 180*C(n,6), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
%F A337897 a(n) = 2*A128766(n) - A000543(n) = A000543(n) - 2*A337896(n) = A128766(n) - A337896(n).
%F A337897 G.f.: x * (1+x) * (1 + 13*x + 62*x^2 + 13*x^3 + x^4) / (1-x)^7.
%t A337897 Table[n^2(7+2n^2+3n^4)/12, {n,30}]
%Y A337897 Cf. A000543 (oriented), A128766 (unoriented), A337896 (chiral).
%Y A337897 Other elements: A331351 (edges), A337898 (cube faces, octahedron vertices).
%Y A337897 Other polyhedra: A006003 (tetrahedron), A337962  (dodecahedron faces, icosahedron vertices), A337960 (icosahedron faces, dodecahedron vertices).
%Y A337897 Row 3 of A337894 (orthoplex faces, orthotope peaks) and A325015 (orthotope vertices, orthoplex facets).
%K A337897 nonn
%O A337897 1,2
%A A337897 _Robert A. Russell_, Sep 28 2020