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%I M4716 #42 Sep 08 2022 08:44:57
%S 0,1,10,57,240,800,2226,5390,11712,23355,43450,76351,127920,205842,
%T 319970,482700,709376,1018725,1433322,1980085,2690800,3602676,4758930,
%U 6209402,8011200,10229375,12937626,16219035,20166832,24885190,30490050
%N Number of inequivalent ways to color faces of a cube using at most n colors.
%C Here inequivalent means under the action of the rotation group of the cube, of order 24, which in its action on the faces has cycle index (x1^6 + 3*x1^2*x2^2 + 6*x1^2*x4 + 6*x2^3 + 8*x3^2)/24.
%C a(n) is also the number of inequivalent colorings of the vertices of a regular octahedron using at most n colors. - _José H. Nieto S._, Jan 19 2012
%C From _Robert A. Russell_, Oct 08 2020: (Start)
%C Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbols for the regular octahedron and cube are {3,4} and {4,3} respectively. They are mutually dual.
%C There are 24 elements in the rotation group of the regular octahedron/cube. 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 Conjugacy Class Count Even Cycle Indices
%C Identity 1 x_1^6
%C Vertex rotation 8 x_3^2
%C Edge rotation 6 x_2^3
%C Small face rotation 6 x_1^2x_4^1
%C Large face rotation 3 x_1^2x_2^2 (End)
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 254 (corrected).
%D N. G. De Bruijn, Polya's theory of counting, in E. F. Beckenbach, ed., Applied Combinatorial Mathematics, Wiley, 1964, pp. 144-184 (see p. 147).
%D M. Gardner, New Mathematical Diversions from Scientific American. Simon and Schuster, NY, 1966, p. 246 (the formula given is incorrect but was corrected in the second printing).
%D J.-P. Delahaye, 'Le miraculeux "lemme de Burnside"','Le coloriage du cube' p. 147 in 'Pour la Science' (French edition of 'Scientific American') No.350 December 2006 Paris.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A047780/b047780.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PolyhedronColoring.html">Polyhedron Coloring</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1)
%F a(n) = (n^6 + 3*n^4 + 12*n^3 + 8*n^2)/24 = n+8*C(n, 2)+30*C(n, 3)+68*C(n, 4)+75*C(n, 5)+30*C(n, 6). Each term of the RHS indicates the number of ways to use n colors to color the cube faces (octahedron vertices) with exactly 1, 2, 3, 4, 5, or 6 colors.
%F G.f.: x*(1+3*x+8*x^2+16*x^3+2*x^4)/(1-x)^7. - _Colin Barker_, Jan 29 2012
%F a(n) = A198833(n) + A093566(n+1) = 2*A198833(n) - A337898(n) = 2*A093566(n+1) + A337898(n). - _Robert A. Russell_, Oct 08 2020
%t CoefficientList[Series[x*(1+3*x+8*x^2+16*x^3+2*x^4)/(1-x)^7,{x,0,33}],x] (* _Vincenzo Librandi_, Apr 27 2012 *)
%o (Magma) [(n^6 + 3*n^4 + 12*n^3 + 8*n^2)/24: n in [1..30]]; // _Vincenzo Librandi_, Apr 27 2012
%Y Cf. A198833 (unoriented), A093566(n+1) (chiral), A337898 (achiral).
%Y Other elements: A060530 (edges), A000543 (cube vertices, octahedron faces).
%Y Cf. A006008 (tetrahedron), A000545 (dodecahedron faces, icosahedron vertices), A054472 (icosahedron faces, dodecahedron vertices).
%Y Row 3 of A325004 (orthoplex vertices, orthotope facets) and A337887 (orthotope faces, orthoplex peaks).
%K nonn,easy
%O 0,3
%A _Jud McCranie_
%E Corrected version of A006550 and A006529.
%E Entry revised by _N. J. A. Sloane_, Jan 03 2005
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