OFFSET
0,2
COMMENTS
This is the number of words of length 2n in the letters x,x^{-1},y,y^{-1} that equal the identity of the Heisenberg group H=<x,y | xz=zx, yz=zy, where z=xyx^{-1}y^{-1}>.
Also, this is the number of closed walks of length 2n on the square lattice enclosing algebraic area 0 [Béguin et al.]. - Andrey Zabolotskiy, Sep 15 2021
LINKS
Cédric Béguin, Alain Valette and Andrzej Zuk, On the spectrum of a random walk on the discrete Heisenberg group and the norm of Harper's operator, Journal of Geometry and Physics, 21 (1997), 337-356.
D. Lind and K, Schmidt, A survey of algebraic actions of the discrete Heisenberg group, arXiv:1502.06243 [math.DS], 2015; Russian Mathematical Surveys, 70:4 (2015), 77-142.
J. Pantone, First 71 terms of the sequence.
FORMULA
Asymptotics: a(n) ~ (1/2) * 16^n * n^(-2).
EXAMPLE
For n=1 the a(1)=4 words are x^{-1}x, xx^{-1}, y^{-1}y, yy^{-1}.
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Igor Pak, Apr 09 2019
STATUS
approved