A287112 1-limiting word of the morphism 0->10, 1->20, 2->0.

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

Offset

1,3

Comments

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

1st: 10

2nd: 2010

3rd: 0102010

4th: 1020100102010

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 = 1.8392867552141611325518525646532866...,

V = U^2 = 3.3829757679062374941227085364...,

W = U^3 = 6.2222625231203986266745611011....

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

From Michel Dekking, Mar 29 2019: (Start)

This sequence is one of the three fixed points of the morphism alpha^3, where alpha is the defining morphism

0->10, 1->20, 2->0.

The other two fixed points are A286998 and A287174.

We have alpha = rho(tau), where tau is the tribonacci morphism in A080843

0->01, 1->02, 2->0,

and rho is the rotation operator.

An eigenvector computation of the incidence matrix of the morphism gives that 0,1, and 2 have frequencies 1/t, 1/t^2 and 1/t^3, where t is the tribonacci constant A058265.

Apparently (u(n)) := A287113. Thus U, the limit of u(n)/n, equals 1/(1/t), the tribonacci constant t. Also, V = A276800, and W = A276801.

(End)

Links

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

Example

1st iterate: 10
4th iterate: 1020100102010
7th iterate:  102010010201020100102010102010010201001020101020100102010201001020101020100102010.

Mathematica

s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 0}, 2 -> 0}] &, {0}, 10]   (* A287112 *)
Flatten[Position[s, 0]] (* A287113 *)
Flatten[Position[s, 1]] (* A287114 *)
Flatten[Position[s, 2]] (* A287115 *)

Crossrefs

Cf. A287113, A287114, A287115, A286998, A287174.

Sequence in context: A178781 A287174 A080843 * A296238 A309475 A221314

Adjacent sequences: A287109 A287110 A287111 * A287113 A287114 A287115

Keyword

nonn,easy

Author

Clark Kimberling, May 22 2017

Status

approved