OFFSET
1,2
COMMENTS
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
LINKS
Francocube forum, [4x4x4] Les maths du 4x4x4
Georges Helm, Rubik's Cube
M. E. Larsen, Rubik's Revenge: The Group Theoretical Solution, Amer. Math. Monthly 92, 381 (1985), DOI:10.2307/2322445.
Christopher Mowla, Math 3900
Robert Munafo, Rubik's Cube and other Cuboid Puzzles
Philippe Picart, Le Rubik's cube
E. Rubik, Rubik Cube Site
Jaap Scherphuis, Puzzle Pages
Xavier Servantie, All about Rubik's cube
Author?, Rubik's Cube
FORMULA
From Andrey Zabolotskiy, Jun 24 2016: (Start)
a(n) = A075152(n)*24 if n is even,
a(n) = A075152(n) if n is odd.
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
EXAMPLE
From Andrey Zabolotskiy, Jun 24 2016 [following Munafo]: (Start)
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)
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Antreas P. Hatzipolakis
EXTENSIONS
a(4) and a(5) corrected and definition clarified by Andrey Zabolotskiy, Jun 24 2016
STATUS
approved