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A128766
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Number of inequivalent n-colorings of the vertices of the 3D cube under full orthogonal group of the cube (of order 48).
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10
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1, 22, 267, 1996, 10375, 41406, 135877, 384112, 966141, 2212750, 4693711, 9340332, 17610307, 31703686, 54839625, 91604416, 148382137, 233880102, 359762131, 541403500, 798782271, 1157522542, 1650105997, 2317268976, 3209603125
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OFFSET
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1,2
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COMMENTS
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The formula was obtained by computing the cycle index of the group of geometric transformations, in 3D space, generated by all possible compositions of the 3 main reflections and the 3 main rotations and their inverses, in any order, with repetition of these geometric transformations allowed.
I assume this refers to colorings of the vertices of the cube. - N. J. A. Sloane, Apr 06 2007
Also the number of ways to color the faces of a regular octahedron with n colors, counting each pair of mirror images as one.
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REFERENCES
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Banks, D. C.; Linton, S. A. & Stockmeyer, P. K. Counting Cases in Substitope Algorithms. IEEE Transactions on Visualization and Computer Graphics, Vol. 10, No. 4, pp. 371-384. 2004.
Perez-Aguila, Ricardo. Enumerating the Configurations in the n-Dimensional Orthogonal Polytopes Through Polya's Counting and A Concise Representation. Proceedings of the 3rd International Conference on Electrical and Electronics Engineering and XII Conference on Electrical Engineering ICEEE and CIE 2006, pp. 63-66.
Polya, G. & Read R. C. Combinatorial Enumeration of Groups, Graphs and Chemical Compounds. Springer-Verlag, 1987.
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LINKS
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FORMULA
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a(n) = (1/48)*(20*n^2 + 21*n^4 + 6*n^6 + n^8).
G.f.: x*(1+x)*(1+12*x+93*x^2+208*x^3+93*x^4+12*x^5+x^6)/(1-x)^9. [Colin Barker, Mar 08 2012]
Cycle Index is (1/48)*(s[1]^8 + 6*s[1]^4*s[2]^2 + 13*s[2]^4 + 8*s[1]^2*s[3]^2 + 12*s[4]^2 + 8*s[2]*s[6]) - Geoffrey Critzer, Mar 31 2013
a(n)=C(n,1)+20C(n,2)+204C(n,3)+1056C(n,4)+2850C(n,5)+4080C(n,6)+2940C(n,7)+840C(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.
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EXAMPLE
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a(2)=22 because there are 22 inequivalent 2-colorings of the 3D cube, including two for which all of the vertices have the same color.
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MATHEMATICA
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A[n_] := (1/48)*(20*n^2 + 21*n^4 + 6*n^6 + n^8)
(*or*)
Drop[Table[CycleIndex[GraphData[{"Hypercube", 3}, "Automorphisms"], s]/.Table[s[i]->n, {i, 1, 8}], {n, 0, 25}], 1] (* Geoffrey Critzer, Mar 31 2013 *)
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CROSSREFS
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Cf. A000543 Number when mirror images are counted separately.
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KEYWORD
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nonn,easy
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AUTHOR
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Ricardo Perez-Aguila (ricardo.perez.aguila(AT)gmail.com), Apr 04 2007
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STATUS
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approved
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