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%I #22 Apr 06 2022 08:29:38
%S 1,4,28,232,2156,21944,240280,2787320,33820044,424925872,5486681368,
%T 72398776344,972270849512,13247921422480,182729003683352,
%U 2546778437385032,35816909974343308,507700854900783784,7246857513425470288,104083322583897779656
%N Cogrowth sequence for the Heisenberg group.
%C 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}>.
%C 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
%H Cédric Béguin, Alain Valette and Andrzej Zuk, <a href="https://doi.org/10.1016/S0393-0440(96)00024-1">On the spectrum of a random walk on the discrete Heisenberg group and the norm of Harper's operator</a>, Journal of Geometry and Physics, 21 (1997), 337-356.
%H D. Lind and K, Schmidt, <a href="https://arxiv.org/abs/1502.06243">A survey of algebraic actions of the discrete Heisenberg group</a>, arXiv:1502.06243 [math.DS], 2015; Russian Mathematical Surveys, 70:4 (2015), 77-142.
%H J. Pantone, <a href="http://www.math.ucla.edu/~pak/hidden/rest/Heisenberg-cogrowth-data.txt">First 71 terms of the sequence</a>.
%F Asymptotics: a(n) ~ (1/2) * 16^n * n^(-2).
%e For n=1 the a(1)=4 words are x^{-1}x, xx^{-1}, y^{-1}y, yy^{-1}.
%Y Related cogrowth sequences: Z A000984, Z^2 A002894, Z^3 A002896, (Z/kZ)^*2 for k = 2..5: A126869, A047098, A107026, A304979, Richard Thompson's group F A246877. The cogrowth sequences for BS(N,N) for N = 2..10 are A229644, A229645, A229646, A229647, A229648, A229649, A229650, A229651, A229652.
%Y Cf. A352838, A178106.
%K nonn,walk
%O 0,2
%A _Igor Pak_, Apr 09 2019
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