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A294781 Growth of the Lamplighter group: number of elements in the Lamplighter group Z ≀ Z of length up to n with respect to the standard generating set {a,t}. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

Table of n, a(n) for n=0..30.

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

FORMULA

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

EXAMPLE

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.

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 *)

CROSSREFS

Cf. A294683. Partial sums of A294782.

Sequence in context: A278464 A186254 A158896 * A110318 A088210 A135344

Adjacent sequences:  A294778 A294779 A294780 * A294782 A294783 A294784

KEYWORD

nonn,easy

AUTHOR

Zoran Sunic, Nov 08 2017

STATUS

approved

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Last modified March 22 04:32 EDT 2019. Contains 321406 sequences. (Running on oeis4.)