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Search: id:A061713
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| A061713 |
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Number of closed walks of length n on a 3 X 3 X 3 Rubik's Cube. |
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+0
1
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| 1, 0, 18, 36, 720, 3600, 42624, 312480, 3148032, 27073152, 261446688, 2407791936, 23168736768, 220481838720, 2137258661472
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Number of n-move sequences on a 3 X 3 X 3 Rubik's Cube (quarter-twists and half-twists count as moves, cf. A060010) that leave the cube unchanged, i.e. closed walks of length n from a fixed vertex on the Cayley graph of the cube with {F, F^(-1), F^2, R, R^(-1), R^2, B, B^(-1), B^2, L, L^(-1), L^2, U, U^(-1), U^2, D, D^(-1), D^2} as the set of generators. Alternatively, the n-th term is equal to the sum of the n-th powers of the eigenvalues of this Cayley graph divided by the order of the Rubik's cube group, ~4.3*10^19 (see A054434).
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EXAMPLE
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There are 18 closed walks of length 2: F*F^(-1), F^2*F^2, F^(-1)*F, R*R^(-1), R^(-1)*R, R^2*R^2 . . ., D*D^(-1), D^(-1)*D, D^2*D^2.
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CROSSREFS
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Cf. A060010, A054434.
Sequence in context: A083211 A023149 A115550 this_sequence A041638 A041636 A041640
Adjacent sequences: A061710 A061711 A061712 this_sequence A061714 A061715 A061716
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KEYWORD
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hard,nonn,nice
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AUTHOR
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Alex Healy (ahealy(AT)fas.harvard.edu), Jun 21 2001
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