A287331 1-limiting word of the morphism 0->10, 1->22, 2->0, starting with 0.

1, 0, 1, 0, 0, 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, 1, 0, 1, 0, 0, 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

Offset: 1

Author

Clark Kimberling, May 23 2017

Keywords

• nonn
• easy

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 1-limiting word is the limit of the words for which the number of iterations is congruent to 1 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}.

Examples

			1st iterate: 10
4th iterate: 1010002210
		

Links

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

Crossrefs

Cf. A287175, A287332, A287333, A287334.

Programs

Mathematica

s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 2}, 2 -> 0}] &, {0}, 10] (* A287331 *)
Flatten[Position[s, 0]] (* A287332 *)
Flatten[Position[s, 1]] (* A287333 *)
Flatten[Position[s, 2]] (* A287334 *)

Extensions

Definition corrected by Georg Fischer, May 27 2021