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A309110 Number of possible permutations of a Pyraminx of size n, including the trivial rotation of the tips.

%I A309110 #25 Feb 24 2025 08:52:36
%S A309110 1,81,75582720,217225462874112000,19228688422470957567836160000000,
%T A309110 52425105093745505455227317179687895040000000000000,
%U A309110 20582183665033346731252760185588627707250626464317440000000000000000000
%N A309110 Number of possible permutations of a Pyraminx of size n, including the trivial rotation of the tips.
%C A309110 Comment rewritten by _Jianing Song_, Feb 23 2025: (Start)
%C A309110 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 A309110 For n >= 3, see the Michael Gottlieb link below for an explanation of the term a(n). (End)
%H A309110 Amiram Eldar, <a href="/A309110/b309110.txt">Table of n, a(n) for n = 1..23</a>
%H A309110 Michael Gottlieb, <a href="https://michael-gottlieb.blogspot.com/2008/05/number-of-positions-of-generalized.html">Notes on Twisty Puzzles</a>
%H A309110 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 A309110 Wikipedia, <a href="https://en.wikipedia.org/wiki/Pyraminx">Pyraminx</a>
%F A309110 a(n) = 272097792 * 369600^(2*n-6) * a(n-3) for n >= 6.
%F A309110 a(n) = 81 * A309109(n) for n >= 2.
%o A309110 (PARI) a(n) = if(n==1, 1, 81 * 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 A309110 Number of possible permutations of: tetrahedron puzzle (without tips: A309109, with tips: this sequence); cube puzzle (A075152); octahedron puzzle (without tips: A309111, with tips: A309112); dodecahedron (A309113).
%K A309110 nonn
%O A309110 1,2
%A A309110 _Jianing Song_, Jul 13 2019