A287108 1-limiting word of the morphism 0->10, 1->21, 2->0.

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

Offset: 1

Author

Clark Kimberling, May 21 2017

Keywords

• nonn
• easy

Comments

Starting with 0, the first 4 iterations of the morphism yield words shown here:

1st: 10

2nd: 2110

3rd: 0212110

4th: 100210212110

The 1-limiting word is the limit of the words for which the number of iterations is congruent to 1 mod 3.

Let u be the sequence of positions of 0, and likewise, v for 1 and w for 2. 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 = 3.079595623491438786010417...,

V = 2.324717957244746025960908...,

W = U + 1 = 4.079595623491438786010417....

If n >=2, then u(n) - u(n-1) is in {1,2,3,4,6}, v(n) - v(n-1) is in {1,2,3,4}, and w(n) - w(n-1) is in {2,3,4,5,7}.

Examples

			The 1st, 4th, and 7th iterates are
10, 100210212110, 10021021211002121102110100210212110211010021211010021100210212110.
		

Links

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

Crossrefs

Cf. A287109, A287110, A287111.

Programs

Mathematica

s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 1}, 2 -> 0}] &, {0}, 10] (* A287108 *)
Flatten[Position[s, 0]] (* A287109 *)
Flatten[Position[s, 1]] (* A287110 *)
Flatten[Position[s, 2]] (* A287111 *)