%I #9 Mar 04 2024 00:14:38

%S 1,3,6,12,22,40,71,123,212,360,607,1017,1693,2807,4635,7629,12524,

%T 20512,33532,54728,89201,145223,236200,383858,623393,1011813,1641441,

%U 2661767,4314821,6992417,11328796,18350552,29719248,48124026,77916923,126140917,204193454

%N Spherical growth function of the Lamplighter group L_2 with respect to the standard generators a, t.

%C Here t and t^{-1} can be thought of as moves left and right, while a=a^{-1} represents the lighting or extinguishing of a lamp.

%H Walter Parry, <a href="https://doi.org/10.1090/S0002-9947-1992-1062874-3">Growth series of some wreath products</a>, Trans. Amer. Math. Soc., Vol. 331 (1992), No. 2, 751-759.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1, 3, 0, -5, -3, 2, 3, 1).

%F G.f.: (1+x)(1-x^2)^2*(1+x+x^2)/((1-x^2-x^3)^2*(1-x-x^2)).

%e Writing L and R for t and t^{-1}, there are 12 elements of the group which can be written as words of length 3, but not more briefly: LLL, LLa, LaL, LaR, aLL, aLa, aRa, aRR, RaL, RaR, RRa, and RRR.

%K nonn

%O 0,2

%A _Andrew Woods_, Jun 08 2017