A287156 2-limiting word of the morphism 0->10, 1->21, 2->0.

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

Offset

1,1

Comments

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

1st: 10

2nd: 2110

3rd: 0212110

4th: 1002010212110

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

Links

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

Example

The 2nd, 5th, and 8th iterates are 2110, 211010021100210212110, 211010021100210212110100210212110021211021101002110021021211002121102110100210212110211010021211010021100210212110.

Mathematica

s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 1}, 2 -> 0}] &, {0}, 11]   (* A287156 *)
Flatten[Position[s, 0]] (* A287157 *)
Flatten[Position[s, 1]] (* A287158 *)
Flatten[Position[s, 2]] (* A287159 *)

Crossrefs

Cf. A287108, A287157, A287158, A287159, A297160.

Sequence in context: A007273 A016319 A342531 * A092510 A117208 A276004

Adjacent sequences: A287153 A287154 A287155 * A287157 A287158 A287159

Keyword

nonn,easy

Author

Clark Kimberling, May 22 2017

Status

approved