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A286998 0-limiting word of the morphism 0->10, 1->20, 2->0.

%I A286998 #48 Jun 17 2025 00:37:05
%S A286998 0,1,0,2,0,1,0,1,0,2,0,1,0,0,1,0,2,0,1,0,2,0,1,0,0,1,0,2,0,1,0,1,0,2,
%T A286998 0,1,0,0,1,0,2,0,1,0,1,0,2,0,1,0,0,1,0,2,0,1,0,2,0,1,0,0,1,0,2,0,1,0,
%U A286998 1,0,2,0,1,0,0,1,0,2,0,1,0,0,1,0,2,0
%N A286998 0-limiting word of the morphism 0->10, 1->20, 2->0.
%C A286998 Starting with 0, the first 5 iterations of the morphism yield words shown here:
%C A286998   1st: 10
%C A286998   2nd: 2010
%C A286998   3rd: 0102010
%C A286998   4th: 1020100102010
%C A286998   5th: 201001020101020100102010
%C A286998 The 2-limiting word is the limit of the words for which the number of iterations is congruent to 2 mod 3.
%C A286998 Let U, V, W be the limits of u(n)/n, v(n)/n, w(n)/n, respectively. Then 1/U + 1/V + 1/W = 1, where
%C A286998   U = 1.8392867552141611325518525646532866..., (A058265)
%C A286998   V = U^2 = 3.3829757679062374941227085364..., (A276800)
%C A286998   W = U^3 = 6.2222625231203986266745611011.... (A276801)
%C A286998 If n >=2, then u(n) - u(n-1) is in {1,2}, v(n) - v(n-1) is in {2,3,4}, and w(n) - w(n-1) is in {4,6,7}.
%C A286998 From _Jiri Hladky_, Aug 29 2021: (Start)
%C A286998 This is also Arnoux-Rauzy word sigma_0 x sigma_1 x sigma_2, where sigmas are defined as:
%C A286998   sigma_0 : 0 -> 0,  1 -> 10, 2 -> 20;
%C A286998   sigma_1 : 0 -> 01, 1 -> 1,  2 -> 21;
%C A286998   sigma_2 : 0 -> 02, 1 -> 12, 2 -> 2.
%C A286998 Fixed point of the morphism 0->0102010, 1->102010, 2->2010, starting from a(1)=0. This definition has the benefit that EACH iteration yields the prefix of the limiting word.
%C A286998 Frequency of letters:
%C A286998   0: 1/t   ~ 54.368% (A192918)
%C A286998   1: 1/t^2 ~ 29.559%
%C A286998   2: 1/t^3 ~ 16.071%
%C A286998 where t is tribonacci constant A058265.
%C A286998 Equals A347290 with a re-mapping of values 1->2, 2->1.
%C A286998 (End)
%H A286998 Jiri Hladky, <a href="/A286998/b286998.txt">Table of n, a(n) for n = 1..20000</a> (terms 1..10000 from Clark Kimberling).
%H A286998 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 A286998 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 A286998 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 A286998 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 A286998 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 A286998 Wikipedia, <a href="https://en.wikipedia.org/wiki/Rauzy_fractal">Rauzy fractal</a>
%H A286998 <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
%e A286998 3rd iterate: 0102010
%e A286998 6th iterate: 01020101020100102010201001020101020100102010
%t A286998 s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 0}, 2 -> 0}] &, {0}, 9] (* A286998 *)
%t A286998 Flatten[Position[s, 0]] (* A286999 *)
%t A286998 Flatten[Position[s, 1]] (* A287000 *)
%t A286998 Flatten[Position[s, 2]] (* A287001 *)
%t A286998 (* Using the 0->0102010, 1->102010, 2->2010 rule: *)
%t A286998 Nest[ Flatten[# /. {0 -> {0, 1, 0, 2, 0, 1, 0}, 1 -> {1, 0, 2, 0, 1, 0}, 2 -> {2, 0, 1, 0}}] &, {0}, 3]
%Y A286998 Cf. A080843, A286999, A287000, A287001, A287112, A287174, A347290 (values 0,2,1).
%K A286998 nonn,easy
%O A286998 1,4
%A A286998 _Clark Kimberling_, May 22 2017