A359797 Cogrowth sequence of the lamplighter group Z_2 wr Z where wr denotes the wreath product.

1, 3, 15, 87, 547, 3623, 24885, 175591, 1265187, 9271167, 68894785, 518053231, 3935274277, 30158804835, 232930956175, 1811476156847, 14174669041427, 111532445963367, 882004732285473, 7006931317108119, 55899039962599777, 447666261592033123

Offset: 0

Author

Andrew Elvey Price, Jan 13 2023

Keywords

• nonn
• walk

Comments

a(n) is the number of words of length 2n in the letters a,t,t^(-1) that equal the identity of the lamplighter group Z_2 wr Z = .

Walks on this group can be seen as operations on an infinite tape of 0's and 1's where each step is either a right shift, left shift or toggles the current element. a(n) is then the number of sequences of 2n such moves which return the tape to the initial position.

Links

• Andrew Elvey Price, Table of n, a(n) for n = 0..500
• Andrew Elvey Price and A. J. Guttmann, Numerical studies of Thompson's group F and related groups, arXiv:1706.07571 [math.GR], 2017.
• D. Revelle, Heat kernel asymptotics on the lamplighter group, Electronic Communications in Probability 8 (2003), 142-154.
• Wikipedia, Lamplighter group

Crossrefs

Spherical growth sequence for this group is A288348.

Cf. A359798.