A287066 - OEIS

%I #12 May 29 2017 23:57:22

%S 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,

%T 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,

%U 1,2,1,2,0,0,1,0,1,1,2,0,1,1,2,1,2,0

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

%C 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

%C U = 3.079595623491438786010417...,

%C V = 2.324717957244746025960908...,

%C W = U + 1 = 4.079595623491438786010417....

%C 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).

%H Clark Kimberling, <a href="/A287066/b287066.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>

%t s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 2}, 2 -> 0}] &, {1}, 10] (* A287066 *)

%t Flatten[Position[s, 0]]  (* A287067 *)

%t Flatten[Position[s, 1]]  (* A287068 *)

%t Flatten[Position[s, 2]]  (* A287069 *)

%Y Cf. A057985, A287067, A287068, A287069.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, May 20 2017