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 A309109 #25 Feb 24 2025 08:52:40 %S A309109 1,1,933120,2681795837952000,237391215092234044047360000000, %T A309109 647223519675870437718855767650467840000000000000, %U A309109 254101032901646255941392101056649724780871931658240000000000000000000 %N A309109 Number of possible permutations of a Pyraminx of size n, disregarding the trivial rotation of the tips. %C A309109 Comment rewritten by _Jianing Song_, Feb 23 2025: (Start) %C A309109 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. %C A309109 For n >= 3, see the Michael Gottlieb link below for an explanation of the term a(n). (End) %H A309109 Amiram Eldar, <a href="/A309109/b309109.txt">Table of n, a(n) for n = 1..23</a> %H A309109 Michael Gottlieb, <a href="https://michael-gottlieb.blogspot.com/2008/05/number-of-positions-of-generalized.html">Notes on Twisty Puzzles</a> %H A309109 Chen Shuang, <a href="https://github.com/cs0x7f/cstimer/wiki/Number-of-states-for-twisty-puzzles">Number of states for twisty puzzles</a>, github. %H A309109 Wikipedia, <a href="https://en.wikipedia.org/wiki/Pyraminx">Pyraminx</a> %F A309109 a(n) = 272097792 * 369600^(2*n-6) * a(n-3) for n >= 6. %F A309109 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). %o A309109 (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)))) %Y A309109 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). %K A309109 nonn %O A309109 1,3 %A A309109 _Jianing Song_, Jul 13 2019