A287566 - OEIS

%I #6 Jul 20 2023 17:56:33

%S 0,2,3,1,3,1,0,2,1,0,2,3,2,3,1,0,1,0,2,3,2,3,1,0,0,2,3,1,3,1,0,2,2,3,

%T 1,0,0,2,3,1,3,1,0,2,1,0,2,3,3,1,0,2,1,0,2,3,2,3,1,0,0,2,3,1,2,3,1,0,

%U 0,2,3,1,3,1,0,2,1,0,2,3,3,1,0,2,1,0

%N Start with 0 and repeatedly substitute 0->0231, 1->2310, 2->3102, 3->1023.

%C This is the fixed point of the morphism 0->0231, 1->2310, 2->3102, 3->1023 starting with 0.  Let t be the (nonperiodic) sequence of positions of 0, and likewise, u for 1, v for 2, and w for 3; then t(n)/n -> 4, u(n)/n -> 4, v(n)/n -> 4,  w(n)/n -> 4.   Also, t(n) + u(n) + v(n) + w(n) = 16*n - 6 for n >= 1.   See A287556 for a guide to related sequences.

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

%F a(n) = 4n - A287568(n) for n >= 1.

%e First three iterations of the morphism:

%e 0231

%e 0231310210232310

%e 0231310210232310102323100231310223100231310210233102102323100231

%t s = Nest[Flatten[# /. {0 -> {0, 2, 3, 1}, 1 -> {2, 3, 2, 0}, 2 -> {3, 1, 0, 2}, 3 -> {1, 0, 2, 3}}] &, {0}, 9];   (* A287566 *)

%t Flatten[Position[s, 0]]; (* A287567 *)

%t Flatten[Position[s, 1]]; (* A287568 *)

%t Flatten[Position[s, 2]]; (* A287569 *)

%t Flatten[Position[s, 3]]; (* A287570 *)

%t Flatten[SubstitutionSystem[{0->{0,2,3,1},1->{2,3,1,0},2->{3,1,0,2},3->{1,0,2,3}},{0},{4}]] (* _Harvey P. Dale_, Jul 20 2023 *)

%Y Cf. A287567, A287568, A287569, A287570.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, May 31 2017