A287002 - OEIS

%I #6 Feb 27 2023 15:17:18

%S 0,1,2,0,1,0,1,2,0,0,1,2,0,1,0,1,0,1,2,0,1,0,1,2,0,0,1,2,0,0,1,2,0,1,

%T 0,1,2,0,0,1,2,0,1,0,1,0,1,2,0,1,0,1,0,1,2,0,1,0,1,2,0,0,1,2,0,1,0,1,

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

%N Start with 0 and repeatedly substitute 0->01, 1->20, 2->1.

%C A fixed point of the morphism 0->01, 1->20, 2->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 = 2.246979603717467061050009768008...,

%C V = 2.801937735804838252472204639014...,

%C W = 5.048917339522305313522214407023...

%C If n >=2, then u(n) - u(n-1) is in {1,2,3}, v(n) - v(n-1) is in {2,3,4}, and w(n) - w(n-1) is in {4,5,7}.

%H Clark Kimberling, <a href="/A287002/b287002.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 -> {2, 0}, 2 -> 1}] &, {0}, 10] (* A287002 *)

%t Flatten[Position[s, 0]]  (* A287003 *)

%t Flatten[Position[s, 1]]  (* A287004 *)

%t Flatten[Position[s, 2]]  (* A287081 *)

%t SubstitutionSystem[{0->{0,1},1->{2,0},2->{1}},{0},{10}][[1]] (* _Harvey P. Dale_, Feb 27 2023 *)

%Y Cf. A287003, A287004, A287081.

%K nonn,easy

%O 1,3

%A _Clark Kimberling_, May 21 2017