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}.

1, 4, 10, 22, 44, 84, 155, 278, 490, 850, 1457, 2474, 4167, 6974, 11609, 19238, 31762, 52274, 85806, 140534, 229735, 374958, 611158, 995016, 1618409, 2630222, 4271663, 6933430, 11248251, 18240668, 29569464, 47920016, 77639264, 125763290, 203680213, 329821130, 534014584

Offset

0,2

Comments

The group is presented by L_2 = <a, t | 1 = a^2 = [a, t^(-k) a t^k], for all k>.

Links

Table of n, a(n) for n=0..36.

Walter Parry, Growth series of some wreath products, Trans. Amer. Math. Soc. 331 (1992), 751-759.

Wikipedia, Lamplighter group

Index entries for linear recurrences with constant coefficients, signature (1, 3, 0, -5, -3, 2, 3, 1).

Formula

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

Example

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.

Mathematica

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 *)
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 *)

Prog

(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

Crossrefs

Partial sums of A288348.

Sequence in context: A052821 A292445 A023628 * A265051 A266375 A004798

Adjacent sequences: A294680 A294681 A294682 * A294684 A294685 A294686

Keyword

nonn,easy

Author

Zoran Sunic, Nov 06 2017

Extensions

More terms from Michel Marcus, Nov 07 2017

Status

approved