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Number of closed walks of length n on a 3 X 3 X 3 Rubik's Cube.
1

%I #6 Jan 16 2022 12:48:14

%S 1,0,18,36,720,3600,42624,312480,3148032,27073152,261446688,

%T 2407791936,23168736768,220481838720,2137258661472

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

%C 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).

%e 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.

%Y Cf. A060010, A054434.

%K hard,nonn,nice

%O 0,3

%A _Alexander D. Healy_, Jun 21 2001