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 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. 9
 1, 21, 201, 1076, 4025, 11901, 29841, 66256, 134001, 251725, 445401, 750036, 1211561, 1888901, 2856225, 4205376, 6048481, 8520741, 11783401, 16026900, 21474201, 28384301, 37055921, 47831376, 61100625, 77305501, 96944121 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. 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.   Conjugacy Class     Count    Odd Cycle Indices   Inversion              1     x_2^4   Vertex rotation*       8     x_2^1x_6^1      Asterisk indicates that the   Edge rotation*         6     x_1^4x_2^2      operation is followed by an   Small face rotation*   3     x_4^2           inversion.   Large face rotation*   6     x_2^4 LINKS FORMULA a(n) = n^2 * (7 + 2*n^2 + 3*n^4) / 12. 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. a(n) = 2*A128766(n) - A000543(n) = A000543(n) - 2*A337896(n) = A128766(n) - A337896(n). G.f.: x * (1+x) * (1 + 13*x + 62*x^2 + 13*x^3 + x^4) / (1-x)^7. MATHEMATICA Table[n^2(7+2n^2+3n^4)/12, {n, 30}] CROSSREFS Cf. A000543 (oriented), A128766 (unoriented), A337896 (chiral). Other elements: A331351 (edges), A337898 (cube faces, octahedron vertices). Other polyhedra: A006003 (tetrahedron), A337962  (dodecahedron faces, icosahedron vertices), A337960 (icosahedron faces, dodecahedron vertices). Row 3 of A337894 (orthoplex faces, orthotope peaks) and A325015 (orthotope vertices, orthoplex facets). Sequence in context: A159857 A221126 A239418 * A341399 A125384 A126542 Adjacent sequences:  A337894 A337895 A337896 * A337898 A337899 A337900 KEYWORD nonn AUTHOR Robert A. Russell, Sep 28 2020 STATUS approved

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Last modified April 22 22:19 EDT 2021. Contains 343197 sequences. (Running on oeis4.)