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 1-5 of 5 results.

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).

A309113 Number of possible permutations of a Megaminx of size 2n+1.

Original entry on oeis.org

1, 100669616553523347122516032313645505168688116411019768627200000000000
Offset: 0

Views

Author

Jianing Song, Jul 13 2019

Keywords

Comments

a(3) has 264 digits and a(4) has 574 digits.
The Megaminx is a dodecahedron-shaped puzzle similar to the Rubik's Cube. The rotational axes of the pieces are perpendicular to the faces. Here only a Megaminx of odd size is considered, see the picture below showing the relationship between the Megaminx of size 2n and 2n+1.

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: A309112); dodecahedron (this sequence).

Programs

  • PARI
    a(n) = if(n, 30! * 20! * 60!^(n^2-1) * 5!^(-12*n^2+12*n) * 2^(28-n) * 3^19, 1)

Formula

a(0) = 1; a(n) = 30! * 20! * 60!^(n^2-1) * 5!^(-12*n^2+12*n) * 2^(28-n) * 3^19 for n > 0.

A330399 Number of possible permutations of a Pyramorphix of size n.

Original entry on oeis.org

1, 136080, 5062877383753728000
Offset: 1

Views

Author

Eder Vanzei, Feb 25 2020

Keywords

Comments

Pyramorphix (must not be confused with Pyraminx) is a tetrahedral edge-turning puzzle, capable of changing shape.

Links

Crossrefs

Cf. A075152, A309109, A309110, A309111, A309112, A309113.
Showing 1-5 of 5 results.

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