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A294782 Spherical growth of the Lamplighter group: number of elements in the Lamplighter group Z ≀ Z of length n with respect to the standard generating set {a,t}. 1
1, 4, 12, 36, 100, 268, 704, 1812, 4600, 11556, 28788, 71252, 175452, 430284, 1051848, 2564708, 6240752, 15161092, 36784284, 89155268, 215911636, 522543436, 1263991824, 3056244212, 7387384808, 17851786148, 43130479748, 104187860340, 251648811212, 607755975820, 1467673342616 (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)^3 (1+x)^3 (1+x^2) / ((1-2x-x^2)(1-x-x^2-x^3)^2).

EXAMPLE

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.

CROSSREFS

Cf. A288348. First differences of A294781.

Sequence in context: A002842 A051041 A192626 * A002906 A191756 A001411

Adjacent sequences:  A294779 A294780 A294781 * A294783 A294784 A294785

KEYWORD

nonn

AUTHOR

Zoran Sunic, Nov 08 2017

STATUS

approved

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Last modified March 26 12:43 EDT 2019. Contains 321497 sequences. (Running on oeis4.)