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Search: id:A154638
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| A154638 |
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a(n) is the number of unique reduced words of length n (i.e. all possible length-reducing cancellations have been applied and words that are equal are counted only once) in the Coxeter group of "Apollonian reflections" in three dimensions. This group has five generators, satisfying (S_i)^2 = (S_i S_j)^3 = I. |
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+0
11
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| 1, 5, 20, 70, 240, 810, 2730, 9180, 30870, 103770, 348840, 1172610, 3941730, 13249980, 44539470, 149717970, 503272440, 1691734410, 5686712730, 19115706780, 64256852070, 215997400170, 726068516040, 2440656636210, 8204191055730, 27578131979580, 92703029288670
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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ABA and BAB are equal, so are counted as only one unique word.
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REFERENCES
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R. L. Graham, J. C. Lagarias, C. L. Mallows, A. R. Wilks and C. Yan, Apollonian circle Packings: Geometry and Group Theory III Higher Dimensions. Discrete and Computational Geometry 35: 37-72 (2006).
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LINKS
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C. L. Mallows, Growing Apollonian packings, J. Integer Sequences 12, article 09.2.1 (2009)
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FORMULA
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Comment from Bill Thurston, Nov 22 2009: There's a handy program (or rather, a constellation of programs) kbmag by Derek Holt et al., which can be used as a package within GAP or as a free-standing program, to try to find an automatic structure for a group. I entered this presentation, and it produced an automatic structure, which implies the growth function is rational: (1 + 2*X + 2*X^2 + X^3)/(1 - 3*X - 3*X^2 + 6*X^3), as reported by kbgrowth.
John Cannon also found this g.f.
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EXAMPLE
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There are 80 square-free words of length 3, but 20 of these fall into 10 equal pairs (e.g. ABA = BAB). So a(3)=70.
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CROSSREFS
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For other sequences relating to the 3-dimensional case, see A154638-A154645.
Sequence in context: A080930 A000343 A005324 this_sequence A054889 A056384 A036683
Adjacent sequences: A154635 A154636 A154637 this_sequence A154639 A154640 A154641
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KEYWORD
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nonn,new
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AUTHOR
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Colin Mallows (colinm(AT)research.avayalabs.com), Jan 13 2009
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EXTENSIONS
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Corrected and extended with g.f. by John Cannon (john(AT)maths.usyd.edu.au) and Bill Thurston (wpt4(AT)cornell.edu), Nov 22 2009
Edited by njas, Nov 22 2009
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