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

1, 1, 933120, 2681795837952000, 237391215092234044047360000000, 647223519675870437718855767650467840000000000000, 254101032901646255941392101056649724780871931658240000000000000000000

Offset: 1

Author

Jianing Song, Jul 13 2019

Keywords

• nonn

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

• Amiram Eldar, Table of n, a(n) for n = 1..23
• Michael Gottlieb, Notes on Twisty Puzzles
• Chen Shuang, Number of states for twisty puzzles, github.
• Wikipedia, Pyraminx

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).