A287267 1-limiting word of the morphism 0->11, 1->02, 2->0.

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

Offset

1,8

Comments

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

1st: 11

2nd: 0202

3rd: 110110

4th: 020211020211

5th: 11011002021101100202

The 1-limiting word is the limit of the words for which the number of iterations is odd.

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 = 2.7692923542386314152404094643350334926...,

V = 2.4498438945029551040577327454145475624...,

W = 4.3344900716222708116779374775820643087...

If n >=2, then u(n) - u(n-1) is in {1,2,3,4,6}, v(n) - v(n-1) is in {1,2,5,6,10}, and w(n) - w(n-1) is in {2,4,8,10,16}.

From Michel Dekking, Oct 09 2019: (Start)

Let u, v and w be the positions of 0, 1 and 2 in this sequence.

The incidence matrix of the defining morphism: 0->10, 1->12, 2->0 has characteristic polynomial chi(u) = u^3 - 2u - 2.

Let Q = [27+ 3*sqrt(57)]^(1/3). Then the real root of the characteristic polynomial chi is lambda := Q/3 + 2/Q.

An eigenvector of lambda is (1, lambda^2-2, -lambda^2+lambda+2).

The Perron-Frobenius Theorem then gives that the asymptotic frequencies f0, f1 and f2 of the letters 0, 1, and 2 are f0 = 1/(1+lambda), f1 = (lambda^2-2)/(1+lambda), and f2 = (-lambda^2+lambda+2)/(1+lambda).

Algebraic expressions for the three constants U, V and W are then given by U = 1/f0, V = 1/f1, W = 1/f2.

(End)

Links

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

Example

3rd iterate: 110110;
5th iterate: 11011002021101100202.

Mathematica

s = Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {0, 2}, 2 -> 0}] &, {0}, 11] (* A287267 *)
Flatten[Position[s, 0]] (* A287268 *)
Flatten[Position[s, 1]] (* A287269 *)
Flatten[Position[s, 2]] (* A287270 *)

Crossrefs

Cf. A287263 (0-limiting word), A287268, A287269, A287270.

Sequence in context: A235168 A060950 A039976 * A317540 A133701 A230417

Adjacent sequences: A287264 A287265 A287266 * A287268 A287269 A287270

Keyword

nonn,easy

Author

Clark Kimberling, May 24 2017

Status

approved