A287320 0-limiting word of the morphism 0->10, 1->22, 2->0

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

Offset

1,3

Comments

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

1st: 10

2nd: 2210

3rd: 002210

4th: 1010002210

5th: 221022101010002210

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 = 2.28537528186132044169516884721360670506...,

V = 3.87512979416277882597397059430967806752...,

W = 3.28537528186132044169516884721360670506...

If n >=2, then u(n) - u(n-1) is in {1,2,4}, v(n) - v(n-1) is in {2,4,6}, and w(n) - w(n-1) is in {1,3,5,9}.

Links

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

Example

3rd iterate: 002210
6th iterate: 002210002210221022101010002210

Mathematica

s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 2}, 2 -> 0}] &, {0}, 12 (* A287320 *)
Flatten[Position[s, 0]] (* A287321 *)
Flatten[Position[s, 1]] (* A287322 *)
Flatten[Position[s, 2]] (* A287323 *)
SubstitutionSystem[{0->{1, 0}, 1->{2, 2}, 2->{0}}, {2}, {10}][[1]] (* Harvey P. Dale, Aug 17 2022 *)

Crossrefs

Cf. A287321, A287322, A287323, A287181. A298200.

Sequence in context: A284092 A293051 A049783 * A210502 A350797 A283988

Adjacent sequences: A287317 A287318 A287319 * A287321 A287322 A287323

Keyword

nonn,easy

Author

Clark Kimberling, May 23 2017

Status

approved