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

Author

Andrew Woods, Jun 08 2017

Keywords

• nonn

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.

Examples

			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.
		

Links

• 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)).