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%I #14 Aug 24 2019 12:28:18
%S 1,4096,8229184826926694400,
%T 102932617000431297816197041062868879933440000000,
%U 23591434633999616817199324204913456263494895712320734212332719660978929664000000000000000
%N Number of possible permutations of a Corner-turning Octahedron of size n, including the trivialrotation of the tips.
%C a(6) has 143 digits and a(7) has 207 digits.
%C The Corner-turning Octahedron is a regular octahedron puzzle in the style of Rubik's Cube. The rotational axes of the pieces are parallel to the lines connecting a pair of opposite vertices. In comparison, the rotational axes of the Face-turning Octahedron are perpendicular to the faces. As a result, the only rotation of the Corner-turning Octahedron of size 2 is the trivial rotation of the tips (it is not the same of the Skewb Diamond, the Face-turning Octahedron of size 2). For n >= 3, see the Michael Gottlieb link below for an explanation of the term a(n).
%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
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