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 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. 8
 1, 10, 55, 200, 560, 1316, 2730, 5160, 9075, 15070, 23881, 36400, 53690, 77000, 107780, 147696, 198645, 262770, 342475, 440440, 559636, 703340, 875150, 1079000, 1319175, 1600326, 1927485, 2306080, 2741950, 3241360 (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 face (octahedron vertex) 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_2^3 Vertex rotation* 8 x_6^1 Asterisk indicates that the Edge rotation* 6 x_1^2x_2^2 operation is followed by an Small face rotation* 6 x_2^1x_4^1 inversion. Large face rotation* 3 x_1^4x_2^1 LINKS Table of n, a(n) for n=1..30. Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1). FORMULA a(n) = n * (n+1) * (n+2) * (3*n^2 - 3*n + 4) / 24. 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. a(n) = 2*A198833(n) - A047780(n) = A047780(n) - 2*A093566(n+1) = A198833(n) - A093566(n+1). G.f.: x * (x + 4*x^2 + 10*x^3) / (1-x)^6. 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 MATHEMATICA Table[n(1+n)(2+n)(4-3n+3n^2)/24, {n, 35}] LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 10, 55, 200, 560, 1316}, 40] (* Harvey P. Dale, Feb 15 2022 *) PROG (PARI) a(n)=n*(n+1)*(n+2)*(3*n^2-3*n+4)/24 \\ Charles R Greathouse IV, Oct 21 2022 CROSSREFS Cf. A047780 (oriented), A198833 (unoriented), A093566(n+1) (chiral). Other elements: A331351 (edges), A337897 (cube vertices/octahedron faces). Other polyhedra: A006003 (simplex), A337962 (dodecahedron faces, icosahedron vertices), A337960 (icosahedron faces, dodecahedron vertices). Row 3 of A325007 (orthotope facets, orthoplex vertices) and A337890 (orthotope faces, orthoplex peaks). Sequence in context: A069155 A107352 A127761 * A357730 A341989 A341070 Adjacent sequences: A337895 A337896 A337897 * A337899 A337900 A337901 KEYWORD nonn,easy AUTHOR Robert A. Russell, Sep 28 2020 STATUS approved

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Last modified September 27 14:34 EDT 2023. Contains 365711 sequences. (Running on oeis4.)