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Spherical growth of the Lamplighter group: number of elements in the Lamplighter group Z wr Z of length n with respect to the standard generating set {a,t}.
2

%I #13 Mar 08 2024 11:48:27

%S 1,4,12,36,100,268,704,1812,4600,11556,28788,71252,175452,430284,

%T 1051848,2564708,6240752,15161092,36784284,89155268,215911636,

%U 522543436,1263991824,3056244212,7387384808,17851786148,43130479748,104187860340,251648811212,607755975820,1467673342616

%N Spherical growth of the Lamplighter group: number of elements in the Lamplighter group Z wr Z of length 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)^3 (1+x)^3 (1+x^2) / ((1-2x-x^2)(1-x-x^2-x^3)^2).

%e a(2)=12, since the elements of length 2 are a^2, at, at^-1, a^-2, a^-1t, a^-1t^-1, ta, ta^-1, t^2, t^-1a, t^-1a^-1, t^-2.

%Y Cf. A288348. First differences of A294781.

%K nonn

%O 0,2

%A _Zoran Sunic_, Nov 08 2017