A061713 Number of closed walks of length n on a 3 X 3 X 3 Rubik's Cube.

1, 0, 18, 36, 720, 3600, 42624, 312480, 3148032, 27073152, 261446688, 2407791936, 23168736768, 220481838720, 2137258661472

Offset: 0

Author

Alexander D. Healy, Jun 21 2001

Keywords

• hard
• nonn
• nice

Comments

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).

Examples

			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.
		

Crossrefs

Cf. A060010, A054434.