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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.
0
1, 5, 20, 80, 300, 1140, 4260
OFFSET
0,2
COMMENTS
ABA and BAB are equal, but are counted as distinct reduced words.
LINKS
R. L. Graham, J. C. Lagarias, C. L. Mallows, Allan Wilks and C. Yan, Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions., Discrete & Computational Geometry, 35 (2006), no. 1, 37-72.
C. L. Mallows, Growing Apollonian Packings, J. Integer Sequences, 12 (2009), article 09.2.1.
EXAMPLE
All 80 squarefree words of length 3 are counted, so a(3) = 80.
CROSSREFS
For other sequences relating to the 3-dimensional case, see A154638-A154645.
Sequence in context: A079820 A275907 A117422 * A214939 A162925 A163316
KEYWORD
more,nonn
AUTHOR
Colin Mallows, Jan 13 2009
STATUS
approved