cp's OEIS Frontend

A337898 Number of achiral colorings of the 6 square faces of a cube or the 6 vertices of a regular octahedron using n or fewer colors.

%I A337898 #23 Aug 07 2023 14:57:58
%S A337898 1,10,55,200,560,1316,2730,5160,9075,15070,23881,36400,53690,77000,
%T A337898 107780,147696,198645,262770,342475,440440,559636,703340,875150,
%U A337898 1079000,1319175,1600326,1927485,2306080,2741950,3241360
%N A337898 Number of achiral colorings of the 6 square faces of a cube or the 6 vertices of a regular octahedron using n or fewer colors.
%C A337898 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 A337898 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 face (octahedron vertex) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
%C A337898   Conjugacy Class     Count    Odd Cycle Indices
%C A337898   Inversion              1     x_2^3
%C A337898   Vertex rotation*       8     x_6^1           Asterisk indicates that the
%C A337898   Edge rotation*         6     x_1^2x_2^2      operation is followed by an
%C A337898   Small face rotation*   6     x_2^1x_4^1      inversion.
%C A337898   Large face rotation*   3     x_1^4x_2^1
%H A337898 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A337898 a(n) = n * (n+1) * (n+2) * (3*n^2 - 3*n + 4) / 24.
%F A337898 a(n) = 1*C(n,1) + 8*C(n,2) + 28*C(n,3) + 36*C(n,4) + 15*C(n,5), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
%F A337898 a(n) = 2*A198833(n) - A047780(n) = A047780(n) - 2*A093566(n+1) = A198833(n) - A093566(n+1).
%F A337898 G.f.: x * (x + 4*x^2 + 10*x^3) / (1-x)^6.
%F A337898 a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - _Wesley Ivan Hurt_, Sep 30 2020
%t A337898 Table[n(1+n)(2+n)(4-3n+3n^2)/24, {n, 35}]
%t A337898 LinearRecurrence[{6,-15,20,-15,6,-1},{1,10,55,200,560,1316},40] (* _Harvey P. Dale_, Feb 15 2022 *)
%o A337898 (PARI) a(n)=n*(n+1)*(n+2)*(3*n^2-3*n+4)/24 \\ _Charles R Greathouse IV_, Oct 21 2022
%Y A337898 Cf. A047780 (oriented), A198833 (unoriented), A093566(n+1) (chiral).
%Y A337898 Other elements: A331351 (edges), A337897 (cube vertices/octahedron faces).
%Y A337898 Other polyhedra: A006003 (simplex), A337962 (dodecahedron faces, icosahedron vertices), A337960 (icosahedron faces, dodecahedron vertices).
%Y A337898 Row 3 of A325007 (orthotope facets, orthoplex vertices) and A337890 (orthotope faces, orthoplex peaks).
%K A337898 nonn,easy
%O A337898 1,2
%A A337898 _Robert A. Russell_, Sep 28 2020