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.

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

Offset: 0

Author

Colin Mallows, Jan 13 2009

Keywords

• more
• nonn

Comments

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

Examples

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

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.

Crossrefs

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