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%I #17 Mar 04 2024 00:16:13
%S 1,5,17,53,153,421,1125,2937,7537,19093,47881,119133,294585,724869,
%T 1776717,4341425,10582177,25743269,62527553,151682821,367594457,
%U 890137893,2154129717,5210373929,12597758737,30449544885,73580024633,177767884973,429416696185,1037172672005,2504846014621
%N Growth of the Lamplighter group: number of elements in the Lamplighter group Z wr Z of length up to n with respect to the standard generating set {a,t}.
%C The group is presented by <a, t | 1 = [a, t^(-k) a t^k], for all k>.
%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. 331 (1992), 751-759.
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (4, -2, -4, -4, 4, 6, 4, 1).
%F G.f.: (1-x)^2 (1+x)^3 (1+x^2) / ((1-2x-x^2)(1-x-x^2-x^3)^2).
%e a(2)=17, since the elements of length up to 2 are 1, a, a^-1, t, t^-1, a^2, at, at^-1, a^-2, a^-1t, a^-1t^-1, ta, ta^-1, t^2, t^-1a, t^-1a^-1, t^-2.
%t CoefficientList[ Series[-((x^2 + 1) (x - 1)^2 (x + 1)^3)/((x^3 + x^2 + x - 1)^2 (x^2 + 2 x - 1)), {x, 0, 27}], x] (* or *)
%t LinearRecurrence[{4, -2, -4, -4, 4, 6, 4, 1}, {1, 5, 17, 53, 153, 421, 1125, 2937}, 28] (* _Robert G. Wilson v_, Aug 08 2018 *)
%Y Cf. A294683. Partial sums of A294782.
%K nonn,easy
%O 0,2
%A _Zoran Sunic_, Nov 08 2017