A057985 Start with 0 and repeatedly substitute: 0->01, 1->12, 2->0.

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, 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, 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

Offset: 1

Author

Clark Kimberling, Oct 30 2000

Keywords

• nonn

Comments

This is the fixed point of the morphism 0->01, 1->12, 2->0 starting with 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. 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). - Clark Kimberling, May 20 2017.

Links

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

Crossrefs

Cf. A287066 (initial term 1 instead of 0).

Programs

Mathematica

t = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 2}, 2 -> {0}}] &, {0}, 10] (* A057985 *)
Flatten[Position[t, 0]] (* A057986 *)
Flatten[Position[t, 1]] (* A057987 *)
Flatten[Position[t, 2]] (* A057988 *)
(* Clark Kimberling, May 13 2013 *)