cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 14 results. Next

A075152 Number of possible permutations of a Rubik cube of size n X n X n.

Original entry on oeis.org

1, 3674160, 43252003274489856000, 7401196841564901869874093974498574336000000000, 282870942277741856536180333107150328293127731985672134721536000000000000000
Offset: 1

Views

Author

Warren Power, Sep 05 2002

Keywords

Comments

More precisely, order of group of n X n X n Rubik cube, under assumptions not-s, not-m, not-i.
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

Crossrefs

See A007458, A054434, A074914, A080656-A080662 for other versions.
Cf. A079761, A079762, A152169 (sums give a(2)), A080601, A080602 (sums give a(3)).

Programs

  • Maple
    f := proc(n) local A,B,C,D,E,F,G; if n mod 2 = 1 then A := (n-1)/2; F := 0; B := 1; C := 1; D := 0; E := (n+1)*(n-3)/4; G := (n-1)*(n-3)/4; else A := n/2; F := 1; B := 1; C := 0; D := 0; E := n*(n-2)/4; G := (n-2)^2/4; 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[n_] := Block[{a, b, c, d, e, f, g}, If[OddQ@ n, a = (n - 1)/2; b = c = 1; d = f = 0; e = (n + 1) (n - 3)/4; g = (n - 1) (n - 3)/4, a = n/2; b = f = 1; c = d = 0; e = n (n - 2)/4; g = (n - 2)^2/4]; Ceiling[(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)]]; Array[f, 10] (* Robert G. Wilson v, May 23 2009 *)
f[1]=1; f[2]=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 *)
f[1]=1;f[n_]:=7!3^6(6*24!!)^(s=Mod[n,2])24!^(r=(n-s)/2-1)(24!/4!^6)^(r(r+s)); Array[f,5] (* Herbert Kociemba, Jul 03 2022 *)
  • Maxima
    A075152(n) := block( if n = 1 then return (1), [a:1,b:1,c:1,d:1,e:1,f:1,g:1], if mod(n, 2) = 1 then (  a : (n-1)/2,  f : 0,  b : 1,  c : 1,  d : 0,  e : (n+1)*(n-3)/4,  g : (n-1)*(n-3)/4 ) else (  a : n/2,  f : 1,   b : 1,   c : 0,   d : 0,   e : n*(n-2)/4, g : (n-2)^2/4  ), return ( (2^a * ((factorial(8)/2)*3^7)^b * ((factorial(12)/2)*2^11)^c * ((4^6)/2)^d * (factorial(24)/2)^e) / (24^f * ((24^6)/2)^g) ) )$ for i:1 thru 27 step 1 do ( sprint(i, A075152(i)), newline() )$ /* Robert Munafo, Nov 12 2014 */
  • PARI
    A075152(n)=ceil(3674160*(11771943321600)^(n%2)*620448401733239439360000^floor((n-2)/2)*(3246670537110000)^floor(((n-2)/2)^2)) \\ Davis Smith, Mar 20 2020

Formula

a(1)=1; a(2)=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
a(n) = ceiling(3674160*11771943321600^(n mod 2)*620448401733239439360000^floor((n - 2)/2)*3246670537110000^floor(((n - 2)/2)^2)). - Davis Smith, Mar 20 2020

Extensions

Entry revised by N. J. A. Sloane, Apr 01 2006
Offset changed to 1 by N. J. A. Sloane, Sep 02 2009

A007458 Order of group of n X n X n Rubik cube.

Original entry on oeis.org

1, 24, 1058158080, 173008013097959424000
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

It would be nice to have a more precise definition of this sequence! - N. J. A. Sloane, Feb 28 2003.
a(2) = A054434(1)*24, a(3) = A054434(2)*12, a(4) = A054434(3)*4. - Andrey Zabolotskiy, Jun 26 2016

References

  • J. H. Conway, personal communication.
  • 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.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

See A054434, A074914, A075152 for other versions.

A074914 Order of group of n X n X n Rubik cube, under assumptions not-s, m, i.

Original entry on oeis.org

1, 3674160, 43252003274489856000, 31180187340244394380451751732775816935095098996162560000000000, 55852096265861522186773299669081144244056150466856272776458775940912440274885530047848906752000000000000000000
Offset: 1

Views

Author

N. J. A. Sloane, Feb 28 2003

Keywords

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.
  • C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 452.
  • 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

Crossrefs

See A007458, A054434, A075152, A080656-A080662 for other versions.

Programs

  • Maple
    f := proc(n) local A,B,C,D,E,F,G; if n mod 2 = 1 then A := (n-1)/2; B := (n-1)/2; C := (n-1)/2; D := 0; E := (n+4)*(n-1)*(n-3)/24; F := 0; G := 0; else A := n/2; B := n/2; C := 0; D := 0; E := n*(n^2-4)/24; 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;

A080656 Order of group of n X n X n Rubik cube, under assumptions not-s, m, not-i.

Original entry on oeis.org

1, 3674160, 43252003274489856000, 707195371192426622240452051915172831683411968000000000, 2582636272886959379162819698174683585918088940054237132144778804568925405184000000000000000
Offset: 1

Views

Author

N. J. A. Sloane, Mar 01 2003

Keywords

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

Crossrefs

See A007458, A054434, A075152, A074914, A080658, A080659, A080660, A080661, A080662 for other versions.

Programs

  • 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 *)

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

A080662 Order of group of n X n X n Rubik cube, under assumptions s, not-m, i.

Original entry on oeis.org

1, 3674160, 88580102706155225088000, 326318176648849198250599213408124182588293120000000000, 25658098810418462614156980952771358874191154069919957663814291417013979423841452032000000000000000000
Offset: 1

Views

Author

N. J. A. Sloane, Mar 01 2003

Keywords

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

Crossrefs

See A007458, A054434, A075152, A074914, A080656-A080661 for other versions.

Programs

  • Maple
    f := proc(n) local A,B,C,D,E,F,G; if n mod 2 = 1 then A := (n-1)/2; F := 0; B := (n-1)/2; C := (n-1)/2; D := (n-1)/2; E := (n+4)*(n-1)*(n-3)/24; G := (n^2-1)*(n-3)/24; else A := n/2; F := 1; B := n/2; C := 0; D := 0; E := n*(n^2-4)/24; G := n*(n-1)*(n-2)/24; 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;

A080661 Order of group of n X n X n Rubik cube, under assumptions s, not-m, not-i.

Original entry on oeis.org

1, 3674160, 88580102706155225088000, 7401196841564901869874093974498574336000000000, 579319689784815322186097322203443872344325595106656531909705728000000000000000
Offset: 1

Views

Author

N. J. A. Sloane, Mar 01 2003

Keywords

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

Crossrefs

See A007458, A054434, A075152, A074914, A080656-A080662 for other versions.

Programs

  • Maple
    f := proc(n) local A,B,C,D,E,F,G; if n mod 2 = 1 then A := (n-1)/2; F := 0; B := 1; C := 1; D := 1; E := (n+1)*(n-3)/4; G := (n-1)*(n-3)/4; else A := n/2; F := 1; B := 1; C := 0; D := 0; E := n*(n-2)/4; G := (n-2)^2/4; 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;

A257401 God's number for a Rubik's cube of size n X n X n (using the half turn metric).

Original entry on oeis.org

0, 11, 20
Offset: 1

Views

Author

Peter Woodward, Apr 21 2015

Keywords

Comments

"God's Number" is the maximum number of turns required to solve any scrambled cube. The "Half turn metric" considers a 90- or 180-degree turn of any side to be a single turn. The number is not known for cubes of size larger than 3 X 3 X 3.
God's number has been proved using a brute-force attack for the 2 X 2 X 2 and 3 X 3 X 3 cubes. For the 4 X 4 X 4 cube, it has been proved only that the lower bound is 31, while the most probable value is considered to be 32; solving this by brute force would require checking all the A075152(4) possible permutations of the "Master Cube". - Marco Ripà, Aug 05 2015

Links

Crossrefs

Cf. A256573 (quarter turn metric), A054434 (possible positions), A075152 (possible permutations).
Cf. A079761, A079762, A080601, A080602.

Formula

From Ben Whitmore, May 31 2021: (Start)
a(n) = Theta(n^2/log(n)) [Demaine et al.].
Conjecture: a(n) ~ (1/4)*log(24!/4!^6) * n^2/log(n).
(End)

A060010 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.

Original entry on oeis.org

1, 12, 312, 10464, 398208, 16323072, 702465024
Offset: 0

Views

Author

Alexander D. Healy, Mar 15 2001

Keywords

Comments

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).

Examples

			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.

Crossrefs

Cf. A054434, A061713, A080602.

A061713 Number of closed walks of length n on a 3 X 3 X 3 Rubik's Cube.

Original entry on oeis.org

1, 0, 18, 36, 720, 3600, 42624, 312480, 3148032, 27073152, 261446688, 2407791936, 23168736768, 220481838720, 2137258661472
Offset: 0

Views

Author

Alexander D. Healy, Jun 21 2001

Keywords

Comments

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).

Examples

			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.

Crossrefs

Cf. A060010, A054434.

A080658 Order of group of n X n X n Rubik cube, under assumptions not-s, not-m, i.

Original entry on oeis.org

1, 3674160, 43252003274489856000, 326318176648849198250599213408124182588293120000000000, 6117367460827460912265057790940131872699535863380422035173008779767508369408000000000000000000
Offset: 1

Views

Author

N. J. A. Sloane, Mar 01 2003

Keywords

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

Crossrefs

See A007458, A054434, A075152, A074914, A080656-A080662 for other versions.

Programs

  • Maple
    f := proc(n) local A,B,C,D,E,F,G; if n mod 2 = 1 then A := (n-1)/2; F := 0; B := (n-1)/2; C := (n-1)/2; D := 0; E := (n+4)*(n-1)*(n-3)/24; G := (n^2-1)*(n-3)/24; else A := n/2; F := 1; B := n/2; C := 0; D := 0; E := n*(n^2-4)/24; G := n*(n-1)*(n-2)/24; 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;
Showing 1-10 of 14 results. Next

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