OFFSET
1,2
COMMENTS
The three possible assumptions considered here are the following:
s (for n odd) indicates that we are working in the "supergroup" and so take account of twists of the face centers.
m (for n > 3) indicates that the pieces are marked so that we take account of the permutation of the identically-colored pieces on a face.
i (for n > 3) indicates that we are working in the theoretical invisible group and solve the pieces on the interior of the cube as well as the exterior. It is assumed that the M and S traits apply to the interior pieces as if they were on the exterior of a smaller cube.
REFERENCES
Dan Hoey, posting to Cube Lovers List, Jun 24, 1987.
Rowley, Chris, The group of the Hungarian magic cube, in Algebraic structures and applications (Nedlands, 1980), pp. 33-43, Lecture Notes in Pure and Appl. Math., 74, Dekker, New York, 1982.
LINKS
Alan Bawden, Cube Lovers Archive, Part 6
FORMULA
a(1)=1; a(2)=7!*3^6; a(3)=8!*3^7*12!*2^10; a(n)=a(n-2)*24!*(24!/2)^(n-3). - Herbert Kociemba, Dec 08 2016
MAPLE
f := proc(n) local A, B, C, D, E, F, G; if n mod 2 = 1 then A := (n-1)/2; B := 1; C := 1; D := 0; E := (n+1)*(n-3)/4; F := 0; G := 0; else A := n/2; B := 1; C := 0; D := 0; E := n*(n-2)/4; F := 1; G := 0; fi; (2^A*((8!/2)*3^7)^B*((12!/2)*2^11)^C*((4^6)/2)^D*(24!/2)^E)/(24^F*((24^6)/2)^G); end;
MATHEMATICA
f[1]=1; f[2]=7!3^6; f[3]=8!3^7 12!2^10; f[n_]:=f[n-2]*24!(24!/2)^(n-3); Array[f, 5] (* Herbert Kociemba, Dec 08 2016 *)
f[1]=1; f[n_]:=7!3^6(6*24!!)^(s=Mod[n, 2])24!^(r=(n-s)/2-1)(24!/2)^(r(r+s)); Array[f, 5] (* Herbert Kociemba, Jul 03 2022 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Mar 01 2003
STATUS
approved