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

1, 1, 2009078326886400, 25130033447370922318407480728239472640000000, 5759627596191312699511553760965199283079808523515804251057792885981184000000000000000

Offset: 1

Author

Jianing Song, Jul 13 2019

Keywords

• nonn

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

• Amiram Eldar, Table of n, a(n) for n = 1..14
• Michael Gottlieb's blogger, Notes on Twisty Puzzles

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).