A154638 a(n) is the number of distinct reduced words of length n in the Coxeter group of "Apollonian reflections" in three dimensions.

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

Offset: 0

Author

Colin Mallows, Jan 13 2009

Keywords

• nonn
• easy

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.

Examples

			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.
		

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).

Crossrefs

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

Programs

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 := GrowthFunction(A); f;
T := PowerSeriesRing(Integers(), 27);
Eltseq(T!f);

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 *)

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

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

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