

A047780


Number of inequivalent ways to color faces of a cube using at most n colors.
(Formerly M4716)


12



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
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OFFSET

0,3


COMMENTS

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.
a(n) is also the number of inequivalent colorings of the vertices of a regular octahedron using at most n colors.  JosÃ© H. Nieto S., Jan 19 2012


REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 254 (corrected).
N. G. De Bruijn, Polya's theory of counting, in E. F. Beckenbach, ed., Applied Combinatorial Mathematics, Wiley, 1964, pp. 144184 (see p. 147).
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' p. 147 in 'Pour la Science' (French edition of 'Scientific American') No.350 December 2006 Paris.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Polyhedron Coloring
Index entries for linear recurrences with constant coefficients, signature (7,21,35,35,21,7,1)


FORMULA

(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.
G.f.: x*(1+3*x+8*x^2+16*x^3+2*x^4)/(1x)^7.  Colin Barker, Jan 29 2012


MATHEMATICA

CoefficientList[Series[x*(1+3*x+8*x^2+16*x^3+2*x^4)/(1x)^7, {x, 0, 33}], x] (* Vincenzo Librandi, Apr 27 2012 *)


PROG

(MAGMA) [(n^6 + 3*n^4 + 12*n^3 + 8*n^2)/24: n in [1..30]]; // Vincenzo Librandi, Apr 27 2012


CROSSREFS

Cf. A000543 (vertices), A060530 (edges).
Cf. A198833 Number when each pair of mirror images is counted as one.
Sequence in context: A067250 A061005 A006550 * A055251 A038733 A004142
Adjacent sequences: A047777 A047778 A047779 * A047781 A047782 A047783


KEYWORD

nonn,easy


AUTHOR

Jud McCranie


EXTENSIONS

Corrected version of A006550 and A006529
Entry revised by N. J. A. Sloane, Jan 03 2005


STATUS

approved



