A309112 - OEIS

%I #18 Feb 21 2025 16:45:02

%S 1,4096,8229184826926694400,

%T 102932617000431297816197041062868879933440000000,

%U 23591434633999616817199324204913456263494895712320734212332719660978929664000000000000000

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

%C a(6) has 143 digits and a(7) has 207 digits.

%C Comment rewritten by _Jianing Song_, Feb 21 2025: (Start)

%C 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.

%C For n >= 3, see the Michael Gottlieb link below for an explanation of the term a(n). (End)

%H Amiram Eldar, <a href="/A309112/b309112.txt">Table of n, a(n) for n = 1..14</a>

%H Michael Gottlieb's blogger, <a href="https://michael-gottlieb.blogspot.com/2008/05/number-of-positions-of-generalized.html">Notes on Twisty Puzzles</a>

%F a(n) = 6^(-16*n+72) * (24!)^(2*n-6) * a(n-3) for n >= 6.

%F a(n) = 4096 * A309111(n) for n >= 2.

%e See the Michael Gottlieb link above.

%o (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)))))

%Y 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).

%K nonn

%O 1,2

%A _Jianing Song_, Jul 13 2019