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

0,2

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 = <a,t | a^2=1, [a,t^(-k)at^k]=1 for all k >.

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.

Sequence in context: A168503 A370184 A089022 * A246538 A132371 A192989

Adjacent sequences: A359794 A359795 A359796 * A359798 A359799 A359800

Keyword

nonn,walk

Author

Andrew Elvey Price, Jan 13 2023

Status

approved