A154639 - OEIS

A154639

a(n) is the number of reduced words of length n (i.e., all possible length-reducing cancellations have been applied) in the generators of the "Apollonian reflection group" in three dimensions. This is a Coxeter group with five generators, satisfying the identities (S_i)^2 = (S_i S_j)^3 = I.

%I #11 Aug 18 2017 16:08:23

%S 1,5,20,80,300,1140,4260

%N a(n) is the number of reduced words of length n (i.e., all possible length-reducing cancellations have been applied) in the generators of the "Apollonian reflection group" in three dimensions. This is a Coxeter group with five generators, satisfying the identities (S_i)^2 = (S_i S_j)^3 = I.

%C ABA and BAB are equal, but are counted as distinct reduced words.

%H R. L. Graham, J. C. Lagarias, C. L. Mallows, Allan Wilks and C. Yan, <a href="http://arxiv.org/abs/math/0010324">Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions.</a>, Discrete & Computational Geometry, 35 (2006), no. 1, 37-72.

%H C. L. Mallows, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Mallows/mallows8.html">Growing Apollonian Packings</a>, J. Integer Sequences, 12 (2009), article 09.2.1.

%e All 80 squarefree words of length 3 are counted, so a(3) = 80.

%Y For other sequences relating to the 3-dimensional case, see A154638-A154645.

%K more,nonn

%O 0,2

%A _Colin Mallows_, Jan 13 2009