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A000543 Number of inequivalent ways to color vertices of a cube using at most n colors. 9
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 (list; graph; refs; listen; history; text; internal format)
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. 144-184 (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)/(1-x)^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)/(1-x)^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

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Last modified October 16 09:29 EDT 2019. Contains 328056 sequences. (Running on oeis4.)