A287066 Start with 1 and repeatedly substitute 0->01, 1->12, 2->0.

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

Offset: 1

Author

Clark Kimberling, May 20 2017

Keywords

• nonn
• easy

Comments

This is the fixed point of the morphism 0->01, 1->12, 2->0 starting with 1. 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}. For n >= 1, the number of terms resulting from n iterations of the morphism is A005251(n+2).

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000
• Index entries for sequences that are fixed points of mappings

Crossrefs

Cf. A057985, A287067, A287068, A287069.

Programs

Mathematica

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 2}, 2 -> 0}] &, {1}, 10] (* A287066 *)
Flatten[Position[s, 0]]  (* A287067 *)
Flatten[Position[s, 1]]  (* A287068 *)
Flatten[Position[s, 2]]  (* A287069 *)