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.
%I A309111 #18 Feb 21 2025 16:44:55 %S A309111 1,1,2009078326886400,25130033447370922318407480728239472640000000, %T A309111 5759627596191312699511553760965199283079808523515804251057792885981184000000000000000 %N A309111 Number of possible permutations of a Corner-Turning Octahedron of size n, disregarding the trivial rotation of the tips. %C A309111 a(6) has 140 digits and a(7) has 203 digits. %C A309111 Comment rewritten by _Jianing Song_, Feb 21 2025: (Start) %C A309111 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 A309111 For n >= 3, see the Michael Gottlieb link below for an explanation of the term a(n). (End) %H A309111 Amiram Eldar, <a href="/A309111/b309111.txt">Table of n, a(n) for n = 1..14</a> %H A309111 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 A309111 a(n) = 6^(-16*n+72) * (24!)^(2*n-6) * a(n-3) for n >= 6. %F A309111 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). %e A309111 See the Michael Gottlieb link above. %o A309111 (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))) %Y A309111 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). %K A309111 nonn %O A309111 1,3 %A A309111 _Jianing Song_, Jul 13 2019