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Number of 2n-move sequences on the 3 X 3 X 3 Rubik's Cube (only quarter-twists count as moves) that leave the cube unchanged.
1

%I #15 Jun 30 2022 08:42:03

%S 1,12,312,10464,398208,16323072,702465024

%N Number of 2n-move sequences on the 3 X 3 X 3 Rubik's Cube (only quarter-twists count as moves) that leave the cube unchanged.

%C I.e., closed walks of length 2n from a fixed vertex on the Cayley graph of the cube with {F, F^(-1), R, R^(-1), B, B^(-1), L, L^(-1) U, U^(-1), D, D^(-1)} 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).

%e There are 12 closed walks of length 2: F*F^(-1), F^(-1)*F, R*R^(-1), R^(-1)*R, ..., D*D^(-1), D^(-1)*D.

%Y Cf. A054434, A061713, A080602.

%K hard,nonn,more

%O 0,2

%A _Alexander D. Healy_, Mar 15 2001