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
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