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 23 results. Next

A080657 Duplicate of A075152.

Original entry on oeis.org

1, 3674160, 43252003274489856000, 7401196841564901869874093974498574336000000000
Offset: 1

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Author

Keywords

A054434 Number of possible positions in an n X n X n Rubik's cube reachable from the starting position.

Original entry on oeis.org

1, 88179840, 43252003274489856000, 177628724197557644876978255387965784064000000000, 282870942277741856536180333107150328293127731985672134721536000000000000000
Offset: 1

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Author

Antreas P. Hatzipolakis

Keywords

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

Examples

			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)

Links

Crossrefs

See A075152, A007458 for other versions.

Programs

  • 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 *)

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(2) = Sum(A080629) = Sum(A080630). (End)
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

Extensions

a(4) and a(5) corrected and definition clarified by Andrey Zabolotskiy, Jun 24 2016

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;

A309109 Number of possible permutations of a Pyraminx of size n, disregarding the trivial rotation of the tips.

Original entry on oeis.org

1, 1, 933120, 2681795837952000, 237391215092234044047360000000, 647223519675870437718855767650467840000000000000, 254101032901646255941392101056649724780871931658240000000000000000000
Offset: 1

Views

Author

Jianing Song, Jul 13 2019

Keywords

Comments

Comment rewritten by Jianing Song, Feb 23 2025: (Start)
The Pyraminx, or the Corner-Turning Tetrahedron, is a regular tetrahedron puzzle in the style of Rubik's Cube. The tetrahedron is cut by 4 groups of n-1 equally-spaced planes, where the planes in each group are perpendicular to one of the 4 faces of the tetrahedron. In comparison, the regular tetrahedron is cut by 3 groups of n-1 equally-spaced planes for the Edge-Turning Tetrahedron of size n, where the planes in each group are parallel to one of the 3 pairs of opposite edges of the tetrahedron. As a result, the Corner-Turning Tetrahedron of size 2 is not the same of the Pyramorphix, the Edge-Turning Tetrahedron of size 2: its only rotations are the trivial rotations of the tips, while the latter is isomorphic to the Rubik's Cube of size 2 as puzzles.
For n >= 3, see the Michael Gottlieb link below for an explanation of the term a(n). (End)

Links

Crossrefs

Number of possible permutations of: tetrahedron puzzle (without tips: this sequence, with tips: A309110); cube puzzle (A075152); octahedron puzzle (without tips: A309111, with tips: A309112); dodecahedron (A309113).

Programs

  • PARI
    a(n) = if(n<=2, 1, 5 * (if(!(n%3), 2^(2*n^2-3*n-1) * 3^(n^2/3+3*n-6) * 1925^(n^2/3-n), 2^(2*n^2-3*n-1) * 3^(n^2/3+3*n-16/3) * 1925^(n^2/3-n-1/3))))

Formula

a(n) = 272097792 * 369600^(2*n-6) * a(n-3) for n >= 6.
a(n) = 5 * 2^(2*n^2-3*n-1) * 3^(n^2/3+3*n-6) * 1925^(n^2/3-n) if 3 divides n, otherwise a(n) = 5 * 2^(2*n^2-3*n-1) * 3^(n^2/3+3*n-16/3) * 1925^(n^2/3-n-1/3).

A309110 Number of possible permutations of a Pyraminx of size n, including the trivial rotation of the tips.

Original entry on oeis.org

1, 81, 75582720, 217225462874112000, 19228688422470957567836160000000, 52425105093745505455227317179687895040000000000000, 20582183665033346731252760185588627707250626464317440000000000000000000
Offset: 1

Views

Author

Jianing Song, Jul 13 2019

Keywords

Comments

Comment rewritten by Jianing Song, Feb 23 2025: (Start)
The Pyraminx, or the Corner-Turning Tetrahedron, is a regular tetrahedron puzzle in the style of Rubik's Cube. The tetrahedron is cut by 4 groups of n-1 equally-spaced planes, where the planes in each group are perpendicular to one of the 4 faces of the tetrahedron. In comparison, the regular tetrahedron is cut by 3 groups of n-1 equally-spaced planes for the Edge-Turning Tetrahedron of size n, where the planes in each group are parallel to one of the 3 pairs of opposite edges of the tetrahedron. As a result, the Corner-Turning Tetrahedron of size 2 is not the same of the Pyramorphix, the Edge-Turning Tetrahedron of size 2: its only rotations are the trivial rotations of the tips, while the latter is isomorphic to the Rubik's Cube of size 2 as puzzles.
For n >= 3, see the Michael Gottlieb link below for an explanation of the term a(n). (End)

Links

Crossrefs

Number of possible permutations of: tetrahedron puzzle (without tips: A309109, with tips: this sequence); cube puzzle (A075152); octahedron puzzle (without tips: A309111, with tips: A309112); dodecahedron (A309113).

Programs

  • PARI
    a(n) = if(n==1, 1, 81 * if(n==2, 1, 5 * (if(!(n%3), 2^(2*n^2-3*n-1) * 3^(n^2/3+3*n-6) * 1925^(n^2/3-n), 2^(2*n^2-3*n-1) * 3^(n^2/3+3*n-16/3) * 1925^(n^2/3-n-1/3)))))

Formula

a(n) = 272097792 * 369600^(2*n-6) * a(n-3) for n >= 6.
a(n) = 81 * A309109(n) for n >= 2.

A309111 Number of possible permutations of a Corner-Turning Octahedron of size n, disregarding the trivial rotation of the tips.

Original entry on oeis.org

1, 1, 2009078326886400, 25130033447370922318407480728239472640000000, 5759627596191312699511553760965199283079808523515804251057792885981184000000000000000
Offset: 1

Views

Author

Jianing Song, Jul 13 2019

Keywords

Comments

a(6) has 140 digits and a(7) has 203 digits.
Comment rewritten by Jianing Song, Feb 21 2025: (Start)
The Corner-Turning Octahedron is a regular octahedron puzzle in the style of Rubik's Cube. The octahedron is cut by 6 groups of n-1 equally-spaced planes not passing through the center, where the planes in each group are perpendicular to one of the 3 lines connecting a pair of opposite vertices of the octahedron. In comparison, the regular octahedron is cut by 4 groups of n-1 equally-spaced planes for the Face-Turning Octahedron of size n, where the planes in each group are parallel to one of the 4 pairs of opposite faces of the octahedron. As a result, the Corner-Turning Octahedron of size 2 is not the same of the Skewb Diamond, the Face-Turning Octahedron of size 2: its only rotations are the trivial rotations of the tips.
For n >= 3, see the Michael Gottlieb link below for an explanation of the term a(n). (End)

Examples

			See the Michael Gottlieb link above.

Links

Crossrefs

Number of possible permutations of: tetrahedron puzzle (without tips: A309109, with tips: A309110); cube puzzle (A075152); octahedron puzzle (without tips: this sequence, with tips: A309112); dodecahedron (A309113).

Programs

  • PARI
    a(n) = if(n<=2, 1, my(A = 258369126400); if(!(n%3), A * 6^(-8*n^2/3+16*n-19) * (24!)^(n^2/3-n), A * 560 * 6^(-8*n^2/3+16*n-43/3) * (24!)^(n^2/3-n-1/3)))

Formula

a(n) = 6^(-16*n+72) * (24!)^(2*n-6) * a(n-3) for n >= 6.
Let A = 258369126400, then for n >= 3: a(n) = A * 6^(-8*n^2/3+16*n-19) * (24!)^(n^2/3-n) if 3 divides n, otherwise a(n) = A * 560 * 6^(-8*n^2+16*n-43/3) * (24!)^(n^2/3-n-1/3).

A309112 Number of possible permutations of a Corner-Turning Octahedron of size n, including the trivial rotation of the tips.

Original entry on oeis.org

1, 4096, 8229184826926694400, 102932617000431297816197041062868879933440000000, 23591434633999616817199324204913456263494895712320734212332719660978929664000000000000000
Offset: 1

Views

Author

Jianing Song, Jul 13 2019

Keywords

Comments

a(6) has 143 digits and a(7) has 207 digits.
Comment rewritten by Jianing Song, Feb 21 2025: (Start)
The Corner-Turning Octahedron is a regular octahedron puzzle in the style of Rubik's Cube. The octahedron is cut by 6 groups of n-1 equally-spaced planes not passing through the center, where the planes in each group are perpendicular to one of the 3 lines connecting a pair of opposite vertices of the octahedron. In comparison, the regular octahedron is cut by 4 groups of n-1 equally-spaced planes for the Face-Turning Octahedron of size n, where the planes in each group are parallel to one of the 4 pairs of opposite faces of the octahedron. As a result, the Corner-Turning Octahedron of size 2 is not the same of the Skewb Diamond, the Face-Turning Octahedron of size 2: its only rotations are the trivial rotations of the tips.
For n >= 3, see the Michael Gottlieb link below for an explanation of the term a(n). (End)

Examples

			See the Michael Gottlieb link above.

Links

Crossrefs

Number of possible permutations of: tetrahedron puzzle (without tips: A309109, with tips: A309110); cube puzzle (A075152); octahedron puzzle (without tips: A309111, with tips: this sequence); dodecahedron (A309113).

Programs

  • PARI
    a(n) = if(n==1, 1, 4096 * (if(n==2, 1, my(A = 258369126400); if(!(n%3), A * 6^(-8*n^2/3+16*n-19) * (24!)^(n^2/3-n), A * 560 * 6^(-8*n^2/3+16*n-43/3) * (24!)^(n^2/3-n-1/3)))))

Formula

a(n) = 6^(-16*n+72) * (24!)^(2*n-6) * a(n-3) for n >= 6.
a(n) = 4096 * A309111(n) for n >= 2.
Showing 1-10 of 23 results. Next

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