A288348 Spherical growth function of the Lamplighter group L_2 with respect to the standard generators a, t.

1, 3, 6, 12, 22, 40, 71, 123, 212, 360, 607, 1017, 1693, 2807, 4635, 7629, 12524, 20512, 33532, 54728, 89201, 145223, 236200, 383858, 623393, 1011813, 1641441, 2661767, 4314821, 6992417, 11328796, 18350552, 29719248, 48124026, 77916923, 126140917, 204193454

Offset

0,2

Comments

Here t and t^{-1} can be thought of as moves left and right, while a=a^{-1} represents the lighting or extinguishing of a lamp.

Links

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

Walter Parry, Growth series of some wreath products, Trans. Amer. Math. Soc., Vol. 331 (1992), No. 2, 751-759.

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

Formula

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

Example

Writing L and R for t and t^{-1}, there are 12 elements of the group which can be written as words of length 3, but not more briefly: LLL, LLa, LaL, LaR, aLL, aLa, aRa, aRR, RaL, RaR, RRa, and RRR.

Prog

(PARI) a(n)=([0, 1, 0, 0, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 0, 1; 1, 3, 2, -3, -5, 0, 3, 1]^n*[1; 3; 6; 12; 22; 40; 71; 123])[1, 1] \\ Charles R Greathouse IV, May 28 2026

Crossrefs

Sequence in context: A246597 A179906 A236913 * A018078 A005404 A249795

Adjacent sequences: A288345 A288346 A288347 * A288349 A288350 A288351

Keyword

nonn,easy,changed

Author

Andrew Woods, Jun 08 2017

Status

approved