A359798 -id:A359798 - cp's OEIS frontend

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

Andrew Elvey Price, Jan 13 2023

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.

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

Spherical growth sequence for this group is A288348.

Cf. A359798.

cp's OEIS Frontend

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

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