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A307468 Cogrowth sequence for the Heisenberg group. 1
1, 4, 28, 232, 2156, 21944, 240280, 2787320, 33820044, 424925872, 5486681368, 72398776344, 972270849512, 13247921422480, 182729003683352, 2546778437385032, 35816909974343308, 507700854900783784, 7246857513425470288, 104083322583897779656 (list; graph; refs; listen; history; text; internal format)



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


Table of n, a(n) for n=0..19.

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.


Asymptotics: a(n) ~ (1/2) * 16^n * n^(-2).


For n=1 the a(1)=4 words are x^{-1}x, xx^{-1}, y^{-1}y, yy^{-1}.


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.

Cf. A352838, A178106.

Sequence in context: A229647 A229646 A229645 * A202824 A046904 A030444

Adjacent sequences: A307465 A307466 A307467 * A307469 A307470 A307471




Igor Pak, Apr 09 2019



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Last modified December 7 09:17 EST 2022. Contains 358654 sequences. (Running on oeis4.)