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A294683 Growth of the Lamplighter group: number of elements in the Lamplighter group L_2 = Z/2Z wr Z of length up to n with respect to the standard generating set {a,t}.

%I A294683 #31 Mar 04 2024 00:13:22
%S A294683 1,4,10,22,44,84,155,278,490,850,1457,2474,4167,6974,11609,19238,
%T A294683 31762,52274,85806,140534,229735,374958,611158,995016,1618409,2630222,
%U A294683 4271663,6933430,11248251,18240668,29569464,47920016,77639264,125763290,203680213,329821130,534014584
%N A294683 Growth of the Lamplighter group: number of elements in the Lamplighter group L_2 = Z/2Z wr Z of length up to n with respect to the standard generating set {a,t}.
%C A294683 The group is presented by L_2 = <a, t | 1 = a^2 = [a, t^(-k) a t^k], for all k>.
%H A294683 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. 331 (1992), 751-759.
%H A294683 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lamplighter_group">Lamplighter group</a>
%H A294683 <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 A294683 G.f.: (1-x)(1+x)^3(1+x+x^2) / ((1-x-x^2)(1-x^2-x^3)^2).
%e A294683 a(2)=10, since the elements of length up to 2 are 1, a, t, t^-1, at, at^-1, ta, t^2, t^-1a, t^-2.
%t A294683 CoefficientList[ Series[((x^2 + x + 1) (x - 1) (x + 1)^3)/((x^3 + x^2 - 1)^2 (x^2 + x - 1)), {x, 0, 36}], x] (* or *)
%t A294683 LinearRecurrence[{1, 3, 0, -5, -3, 2, 3, 1}, {1, 4, 10, 22, 44, 84, 155, 278}, 37] (* _Robert G. Wilson v_, Aug 08 2018 *)
%o A294683 (PARI) Vec((1-x)*(1+x)^3*(1+x+x^2)/((1-x-x^2)*(1-x^2-x^3)^2) + O(x^40)) \\ _Michel Marcus_, Nov 07 2017
%Y A294683 Partial sums of A288348.
%K A294683 nonn,easy
%O A294683 0,2
%A A294683 _Zoran Sunic_, Nov 06 2017
%E A294683 More terms from _Michel Marcus_, Nov 07 2017