A287072 Start with 0 and repeatedly substitute 0->01, 1->21, 2->0.

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

Offset

1,3

Comments

A fixed point of the morphism 0->01, 1->21, 2->0. 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....

Since the morphism 0->01, 1->21, 2->0 is the time reversal of the morphism 0->10, 1->12 2->0, which has fixed point A287104, in particular the incidence matrices of these two morphisms are equal. Thus the algebraic expressions found for U, V and W in A287104 do also apply to the U, V and W above. - Michel Dekking, Sep 15 2019

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

Links

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

James Currie, Pascal Ochem, Narad Rampersad, and Jeffrey Shallit, Properties of a Ternary Infinite Word, arXiv:2206.01776 [cs.DM], 2022.

James Currie, Pascal Ochem, Narad Rampersad, and Jeffrey Shallit, Complement Avoidance in Binary Words, arXiv:2209.09598 [math.CO], 2022.

Index entries for sequences that are fixed points of mappings

Mathematica

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {2, 1}, 2 -> 0}] &, {0}, 10] (* A287072 *)
Flatten[Position[s, 0]] (* A287073 *)
Flatten[Position[s, 1]] (* A287074 *)
Flatten[Position[s, 2]] (* A287075 *)

Crossrefs

Cf. A057985, A287073, A287074, A287075, A287104.

Sequence in context: A291336 A208664 A030200 * A095734 A137269 A343348

Adjacent sequences: A287069 A287070 A287071 * A287073 A287074 A287075

Keyword

nonn,easy

Author

Clark Kimberling, May 21 2017

Status

approved