

A000543


Number of inequivalent ways to color vertices of a cube using at most n colors.


8



0, 1, 23, 333, 2916, 16725, 70911, 241913, 701968, 1798281, 4173775, 8942021, 17930628, 34009053, 61518471, 106823025, 179003456, 290715793, 459239463, 707740861, 1066780100, 1576090341, 2286660783, 3263156073, 4586706576
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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 vertices has cycle index (x1^8 + 9*x2^4 + 6*x4^2 + 8*x1^2*x3^2)/24.
Also the number of ways to color the faces of a regular octahedron with n colors, counting mirror images separately.


REFERENCES

N. G. De Bruijn, Polya's theory of counting, in E. F. Beckenbach, ed., Applied Combinatorial Mathematics, Wiley, 1964, pp. 144184 (see p. 147).


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 (9,36,84,126,126,84,36,9,1).


FORMULA

a(n) = (1/24)*n^2*(n^6+17*n^2+6). (Replace all x_i's in the cycle index with n.)
G.f.: x*(1+x)*(1+13*x+149*x^2+514*x^3+149*x^4+13*x^5+x^6)/(1x)^9.  Colin Barker, Jan 29 2012
a(n) = C(n,1) + 21*C(n,2) + 267*C(n,3) + 1718*C(n,4) + 5250*C(n,5) + 7980*C(n,6) + 5880*C(n,7) + 1680*C(n,8). Each term indicates the number of ways to use n colors to color the cube vertices (octahedron faces) with exactly 1, 2, 3, 4, 5, 6, 7, or 8 colors.


MAPLE

f:= n>(1/24)*n^2*(n^6+17*n^2+6);


MATHEMATICA

CoefficientList[Series[x*(1+x)*(1+13*x+149*x^2+514*x^3+149*x^4+13*x^5+x^6)/(1x)^9, {x, 0, 30}], x] (* Vincenzo Librandi, Apr 15 2012 *)


PROG

(MAGMA) [(1/24)*n^2*(n^6+17*n^2+6): n in [0..30]]; // Vincenzo Librandi, Apr 15 2012


CROSSREFS

Cf. A047780 (faces), A060530 (edges).
Cf. A006550.
Cf. A128766. Number when each pair of mirror images is counted as one.
Sequence in context: A022747 A270498 A260727 * A220648 A243422 A028110
Adjacent sequences: A000540 A000541 A000542 * A000544 A000545 A000546


KEYWORD

nonn,easy


AUTHOR

Clint. C. Williams (Clintwill(AT)aol.com)


EXTENSIONS

Entry revised by N. J. A. Sloane, Jan 03 2005


STATUS

approved



