%I #32 Sep 26 2021 16:15:53
%S 0,2,0,1,0,2,0,2,0,1,0,2,0,0,2,0,1,0,2,0,1,0,2,0,0,2,0,1,0,2,0,2,0,1,
%T 0,2,0,0,2,0,1,0,2,0,2,0,1,0,2,0,0,2,0,1,0,2,0,1,0,2,0,0,2,0,1,0,2,0,
%U 2,0,1,0,2,0,0
%N Arnoux-Rauzy word sigma_0 x sigma_2 x sigma_1. Fixed point of the morphism 0-> 0201020, 1->1020, 2->201020 starting from a(1)=0.
%C Arnoux-Rauzy word sigma_0 x sigma_2 x sigma_1, where sigmas are defined as:
%C sigma_0 : 0 -> 0, 1 -> 10, 2 -> 20;
%C sigma_1 : 0 -> 01, 1 -> 1, 2 -> 21;
%C sigma_2 : 0 -> 02, 1 -> 12, 2 -> 2.
%C Fixed point of the morphism 0->0201020, 1->1020, 2->201020 starting from a(1)=0.
%C Frequency of letters:
%C 0: 1/t ~ 54.368% (A192918)
%C 1: 1/t^3 ~ 16.071%
%C 2: 1/t^2 ~ 29.559%
%C where t is tribonacci constant A058265.
%C Equals A286998 with a re-mapping of values 1->2, 2->1.
%H L. Balková, M. Bucci, A. De Luca, J. Hladký and S. Puzynina, <a href="https://doi.org/10.1016/j.tcs.2016.07.042">Aperiodic Pseudorandom Number Generators Based on Infinite Words</a>, Theoret. Comput. Sci. 647 (2016), 85-100.
%H Julien Cassaigne, Sebastien Ferenczi, and Luca Q. Zamboni, <a href="http://www.numdam.org/item?id=AIF_2000__50_4_1265_0">Imbalances in Arnoux-Rauzy sequences</a>, Annales de l'institut Fourier, 50 (2000), 1265-1276.
%H D. Damanik and L. Q. Zamboni, <a href="https://arxiv.org/abs/math/0208137">Arnoux-Rauzy subshifts: linear recurrence, powers and palindromes</a>, arXiv:math/0208137 [math.CO], 2002.
%H J. Patera, <a href="http://sts-karelia09.jinr.ru/publish/Pepan/v-33-7/20.pdf">Generating the Fibonacci chain in O(log n) space and O(n) time</a> (2003)
%H Gérard Rauzy, <a href="https://doi.org/10.24033/bsmf.1957">Nombres algébriques et substitutions</a>, Bull. Soc. Math. France 110.2 (1982): 147-178.
%H <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
%t Nest[ Flatten[# /. {0 -> {0, 2, 0, 1, 0, 2, 0}, 1 -> {1, 0, 2, 0}, 2 -> {2, 0, 1, 0, 2, 0}] &, {0}, 8]
%Y Cf. A080843 A286998 (values 0,2,1), A058265.
%K nonn
%O 1,2
%A _Jiri Hladky_, Aug 29 2021