A287360 0-limiting word of the morphism 0->11, 1->21, 2->0.

0, 2, 1, 0, 2, 1, 2, 1, 2, 1, 1, 1, 0, 2, 1, 0, 2, 1, 0, 2, 1, 2, 1, 2, 1, 1, 1, 0, 2, 1, 2, 1, 2, 1, 1, 1, 0, 2, 1, 2, 1, 2, 1, 1, 1, 0, 2, 1, 1, 1, 0, 2, 1, 1, 1, 0, 2, 1, 0, 2, 1, 0, 2, 1, 2, 1, 2, 1, 1, 1, 0, 2, 1, 0, 2, 1, 0, 2, 1, 2, 1, 2, 1, 1, 1, 0

Offset

1,2

Comments

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

1st: 11

2nd: 2121

3rd: 021021

4th: 1102111021

5th: 212111021212111021

The 0-limiting word is the limit of the words for which the number of iterations is congruent to 0 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 = 5.5707505637226408833903376944272134...,

V = 1.9375648970813894129869852971548390...,

W = 3.2853752818613204416951688472136067...

If n >=2, then u(n) - u(n-1) is in {3,5,9}, v(n) - v(n-1) is in {1,2,3}, and w(n) - w(n-1) is in {2,3,5}.

Links

Clark Kimberling, Table of n, a(n) for n = 1..10000

Example

3rd iterate: 021021
6th iterate: 021021212111021021021212111021

Mathematica

s = Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {2, 1}, 2 -> 0}] &, {0}, 12] (* A287360 *)
Flatten[Position[s, 0]] (* A287361 *)
Flatten[Position[s, 1]] (* A287362 *)
Flatten[Position[s, 2]] (* A287363 *)
SubstitutionSystem[{0->{1, 1}, 1->{2, 1}, 2->{0}}, {0}, {9}][[1]] (* Harvey P. Dale, Dec 04 2022 *)

Crossrefs

Cf. A287361, A287362, A287363, A287364 (1-limiting word);, A287368 (2-limiting word).

Sequence in context: A190427 A287108 A333948 * A035443 A180430 A246369

Adjacent sequences: A287357 A287358 A287359 * A287361 A287362 A287363

Keyword

nonn,easy

Author

Clark Kimberling, May 24 2017

Status

approved