A294781 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}.
1, 5, 17, 53, 153, 421, 1125, 2937, 7537, 19093, 47881, 119133, 294585, 724869, 1776717, 4341425, 10582177, 25743269, 62527553, 151682821, 367594457, 890137893, 2154129717, 5210373929, 12597758737, 30449544885, 73580024633, 177767884973, 429416696185, 1037172672005, 2504846014621
Offset: 0
Examples
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.
Links
- Walter Parry, Growth series of some wreath products, Trans. Amer. Math. Soc. 331 (1992), 751-759.
- Index entries for linear recurrences with constant coefficients, signature (4, -2, -4, -4, 4, 6, 4, 1).
Programs
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Mathematica
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 *) 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 *)
Formula
G.f.: (1-x)^2 (1+x)^3 (1+x^2) / ((1-2x-x^2)(1-x-x^2-x^3)^2).
Comments