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A154638 a(n) is the number of distinct reduced words of length n in the Coxeter group of "Apollonian reflections" in three dimensions. 2316
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; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Definition means that all possible length-reducing 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], 2001-2005.

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: 37-72 (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 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. - 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 3-dimensional case, see A154638-A154645.

Sequence in context: A000343 A005324 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

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Last modified August 24 00:29 EDT 2017. Contains 291052 sequences.