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Search: id:A128766
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| A128766 |
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Number of inequivalent n-colorings of the 3D cube under full orthogonal group of the cube (of order 48). |
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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
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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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 (njas(AT)research.att.com), Apr 06 2007
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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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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, Orthogonal Polytopes: Study and Application, PhD Thesis. Universidad de las Americas, Puebla. November, 2006.
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.
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FORMULA
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a(n) = (1/48)*(20*n^2 + 21*n^4 + 6*n^6 + n^8)
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EXAMPLE
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a(3)=22 because there are 22 inequivalent 2-colorings of the 3D cube.
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MATHEMATICA
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A[n_] := (1/48)*(20*n^2 + 21*n^4 + 6*n^6 + n^8)
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CROSSREFS
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Cf. A000616, A002817.
Sequence in context: A143479 A004412 A055756 this_sequence A125434 A023020 A022650
Adjacent sequences: A128763 A128764 A128765 this_sequence A128767 A128768 A128769
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KEYWORD
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nonn
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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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