

A154638


a(n) is the number of distinct reduced words of length n in the Coxeter group of "Apollonian reflections" in three dimensions.


2315



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
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OFFSET

0,2


COMMENTS

Definition means that all possible lengthreducing cancellations have been applied and words that are equal are counted only once.
This group has five generators, satisfying (S_i)^2 = (S_i S_j)^3 = I.
ABA and BAB are equal, so are counted as only one distinct word.


LINKS

K. Brockhaus, Table of n, a(n) for n = 0..1000
R. L. Graham, J. C. Lagarias, C. L. Mallows, Allan Wilks and C. Yan, Apollonian circle Packings: Geometry and Group Theory III Higher Dimensions, arXiv:math/0010324 [math.MG], 20012005.
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 and Computational Geometry 35: 3772 (2006).
C. L. Mallows, Growing Apollonian packings, J. Integer Sequences 12, article 09.2.1 (2009)
Index entries for linear recurrences with constant coefficients, signature (3, 3, 6).


FORMULA

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 freestanding 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.  William P. Thurston, Nov 22 2009
Recurrence: for n >= 1, a(n+3) = 3*a(n+2) + 3*a(n+1)  6*a(n) with a(0..3)={1,5,20,70}.  Zak Seidov, Dec 07 2009


EXAMPLE

There are 80 squarefree words of length 3, but 20 of these fall into 10 equal pairs (e.g., ABA = BAB). So a(3)=70.


MATHEMATICA

CoefficientList[Series[(z^3 + 2 z^2 + 2 z + 1)/(6 z^3  3 z^2  3 z + 1), {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 24 2011 *)
Join[{1}, LinearRecurrence[{3, 3, 6}, {5, 20, 70}, 30]] (* Harvey P. Dale, Nov 16 2011 *)


PROG

(Magma) /* gives growth function and terms of sequence  from Klaus Brockhaus, Feb 13 2010 */
G := Group< s1, s2, s3, s4, s5  [ s1^2, s2^2, s3^2, s4^2, s5^2, (s1*s2)^3, (s1*s3)^3, (s1*s4)^3, (s1*s5)^3, (s2*s3)^3, (s2*s4)^3, (s2*s5)^3, (s3*s4)^3, (s3*s5)^3, (s4*s5)^3 ] >;
A := AutomaticGroup(G);
f<x> := GrowthFunction(A); f;
T := PowerSeriesRing(Integers(), 27);
Eltseq(T!f);
(PARI) a(n)=if(n, ([0, 1, 0; 0, 0, 1; 6, 3, 3]^n*[5/6; 5; 20])[1, 1], 1) \\ Charles R Greathouse IV, Jun 11 2015


CROSSREFS

For other sequences relating to the 3dimensional case, see A154638A154645.
Sequence in context: A005324 A304011 A243869 * A054889 A056384 A036683
Adjacent sequences: A154635 A154636 A154637 * A154639 A154640 A154641


KEYWORD

nonn,easy


AUTHOR

Colin Mallows, Jan 13 2009


EXTENSIONS

Corrected and extended with g.f. by John Cannon and William P. Thurston, Nov 22 2009
Edited by N. J. A. Sloane, Nov 22 2009


STATUS

approved



