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. The first 20 coefficients are {1, 5, 20, 70, 240, 810, 2730, 9180, 30870, 103770, 348840, 1172610, 3941730, 13249980, 44539470, 149717970, 503272440, 1691734410, 5686712730, 19115706780, 64256852070} Bill Thurston On Nov 21, 2009, at 10:11 PM, N. J. A. Sloane wrote:
To Math Fun, Seq Fan:
I ran into my distinguished (former) colleague Colin Mallows yesterday, and he said that it would be nice if someone would extend his sequence A154638.
Take the infinite reflection group with 5 generators S_1, ..., S_5, satisfying (S_i)^2 = (S_i S_j)^3 = I, and let a(n) be number of distinct elements whose minimal representation as a product of generators has length n:
1, 5, 20, 70, 240, 780, 2730, .. (for n>= 0)
Can anybody help extend this sequence? Is there a generating function? What about other groups? This might be a gaping hole in the OEIS! There must be a huge literature on this problem.
The books that I know about that might be related, Lyndom & Schupp, Combinatorial Group Theory, Magnus, Karrass, Solitar, same title, Johnson, Presentations of Groups, aren't exactly full of sequences, as far as I can tell.
Can some expert please help?
Neil
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