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A054434
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Number of possible positions in an n X n X n Rubik's cube reachable from the starting position.
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14
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OFFSET
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1,2
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COMMENTS
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The sequence counts possible positions of the Rubik's cube considering the positions which are related through rotations of the cube as a whole (there are 24 of those) as distinct. At odd n, the orientation of the cube as a whole is usually considered fixed by the central squares of each face (i. e., the cube as a whole cannot be rotated) so there is a difference compared to A075152 only in the case of even n. - Andrey Zabolotskiy, Jun 07 2016
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LINKS
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FORMULA
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a(1)=1; a(2)=24*7!*3^6; a(3)=8!*3^7*12!*2^10; a(n)=a(n-2)*24^6*(24!/24^6)^(n-2). - Herbert Kociemba, Dec 08 2016
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EXAMPLE
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a(4) = 8! * 3^7 * 24! * 24! / 4!^6 is constituted by:
8! permutation of corners
× (12*2)! permutation of edges
× (6*4)! permutation of centers
× 1 (combination of permutations must be even, but we can achieve what appears to be an odd permutation of the other pieces in the cube by "hiding" a transposition within the indistinguishable pieces of one color)
× 3^8 orientations of corners
/ 3 total orientation of corners must be zero
× 1 (orientations of edges and centers are determined by their position)
/ 4!^6 the four center pieces of each color are indistinguishable
(End)
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MATHEMATICA
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f[1]=1; f[2]=24*7!3^6; f[3]=8!3^7 12!2^10; f[n_]:=f[n-2]*24^6*(24!/24^6)^(n-2); Table[f[n], {n, 1, 10}] (* Herbert Kociemba, Dec 08 2016 *)
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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Antreas P. Hatzipolakis
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EXTENSIONS
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STATUS
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approved
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