A287566 Start with 0 and repeatedly substitute 0->0231, 1->2310, 2->3102, 3->1023.

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, 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, 0, 2, 3, 1, 3, 1, 0, 2, 1, 0, 2, 3, 3, 1, 0, 2, 1, 0

Offset: 1

Author

Clark Kimberling, May 31 2017

Keywords

• nonn
• easy

Comments

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.

Examples

			First three iterations of the morphism:
0231
0231310210232310
0231310210232310102323100231310223100231310210233102102323100231
		

Links

• Index entries for sequences that are fixed points of mappings

Crossrefs

Cf. A287567, A287568, A287569, A287570.

Programs

Mathematica

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 *)
Flatten[Position[s, 0]]; (* A287567 *)
Flatten[Position[s, 1]]; (* A287568 *)
Flatten[Position[s, 2]]; (* A287569 *)
Flatten[Position[s, 3]]; (* A287570 *)
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 *)

Formula

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