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Search: id:A047780
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| A047780 |
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Number of inequivalent ways to color faces of a cube using at most n colors.
(Formerly M4716)
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+0
7
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| 0, 1, 10, 57, 240, 800, 2226, 5390, 11712, 23355, 43450, 76351, 127920, 205842, 319970, 482700, 709376, 1018725, 1433322, 1980085, 2690800, 3602676, 4758930, 6209402, 8011200, 10229375, 12937626, 16219035, 20166832, 24885190, 30490050
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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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.
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REFERENCES
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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).
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 254 (corrected).
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).
J.-P. Delahaye, 'Le miraculeux "lemme de Burnside"','Le coloriage du cube' pp 147 in 'Pour la Science' (French edition of 'Scientific American') No.350 December 2006 Paris.
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LINKS
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Eric Weisstein's World of Mathematics, Polyhedron Coloring
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FORMULA
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(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).
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CROSSREFS
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Cf. A000543 (vertices), A060530 (edges).
Adjacent sequences: A047777 A047778 A047779 this_sequence A047781 A047782 A047783
Sequence in context: A067250 A061005 A006550 this_sequence A055251 A038733 A004142
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KEYWORD
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nonn
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AUTHOR
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Jud McCranie (j.mccranie(AT)comcast.net)
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EXTENSIONS
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Corrected version of A006550 and A006529.
Entry revised by njas, Jan 03 2005
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