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

Offset

1,4

Comments

Starting with 0, the first 5 iterations of the morphism yield words shown here:

1st: 10

2nd: 2010

3rd: 0102010

4th: 1020100102010

5th: 201001020101020100102010

The 2-limiting word is the limit of the words for which the number of iterations is congruent to 2 mod 3.

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

U = 1.8392867552141611325518525646532866..., (A058265)

V = U^2 = 3.3829757679062374941227085364..., (A276800)

W = U^3 = 6.2222625231203986266745611011.... (A276801)

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}.

From Jiri Hladky, Aug 29 2021: (Start)

This is also Arnoux-Rauzy word sigma_0 x sigma_1 x sigma_2, where sigmas are defined as:

sigma_0 : 0 -> 0, 1 -> 10, 2 -> 20;

sigma_1 : 0 -> 01, 1 -> 1, 2 -> 21;

sigma_2 : 0 -> 02, 1 -> 12, 2 -> 2.

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.

Frequency of letters:

0: 1/t ~ 54.368% (A192918)

1: 1/t^2 ~ 29.559%

2: 1/t^3 ~ 16.071%

where t is tribonacci constant A058265.

Equals A347290 with a re-mapping of values 1->2, 2->1.

(End)

Links

Jiri Hladky, Table of n, a(n) for n = 1..20000 (terms 1..10000 from Clark Kimberling).

L. Balková, M. Bucci, A. De Luca, J. Hladký, and S. Puzynina: Aperiodic Pseudorandom Number Generators Based on Infinite Words, Theoret. Comput. Sci. 647 (2016), 85-100.

Julien Cassaigne, Sebastien Ferenczi, and Luca Q. Zamboni, Imbalances in Arnoux-Rauzy sequences, Annales de l'institut Fourier, 50 (2000), 1265-1276.

D. Damanik and L. Q. Zamboni, Arnoux-Rauzy subshifts: linear recurrence, powers and palindromes, arXiv:math/0208137 [math.CO], 2002.

J. Patera, GENERATING THE FIBONACCI CHAIN IN O (log n) SPACE AND O (n) TIME (2003)

Gérard Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France 110.2 (1982): 147-178.

Wikipedia, Rauzy fractal

Index entries for sequences that are fixed points of mappings

Example

3rd iterate: 0102010
6th iterate: 01020101020100102010201001020101020100102010

Mathematica

s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 0}, 2 -> 0}] &, {0}, 9] (* A286998 *)
Flatten[Position[s, 0]] (* A286999 *)
Flatten[Position[s, 1]] (* A287000 *)
Flatten[Position[s, 2]] (* A287001 *)
Using the 0->0102010, 1->102010, 2->2010 rule:
Nest[ Flatten[# /. {0 -> {0, 1, 0, 2, 0, 1, 0}, 1 -> {1, 0, 2, 0, 1, 0}, 2 -> {2, 0, 1, 0}] &, {0}, 8]

Crossrefs

Cf. A080843, A286999, A287000, A287001, A287112, A287174, A347290 (values 0,2,1).

Sequence in context: A039975 A358679 A016253 * A097796 A117188 A341514

Adjacent sequences: A286995 A286996 A286997 * A286999 A287000 A287001

Keyword

nonn,easy

Author

Clark Kimberling, May 22 2017

Status

approved