A287561 Start with 0 and repeatedly substitute 0->0213, 1->2130, 2->1302, 3->3021.

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

Offset: 1

Author

Clark Kimberling, May 31 2017

Keywords

• nonn
• easy

Comments

This is the fixed point of the morphism 0->0213, 1->2130, 2->1302, 3->3021 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:
0213
0213130221303021
0213130221303021213030210213130213022130302102133021021313022130
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..20000
• Index entries for sequences that are fixed points of mappings

Crossrefs

Cf. A287562, A287563, A287564, A287565.

Programs

Mathematica

s = Nest[Flatten[# /. {0 -> {0, 2, 1, 3}, 1 -> {2, 1, 3, 0}, 2 -> {1, 3, 0, 2}, 3 -> {3, 0, 2, 1}}] &, {0}, 9];   (* A287561 *)
Flatten[Position[s, 0]]; (* A287562 *)
Flatten[Position[s, 1]]; (* A287563 *)
Flatten[Position[s, 2]]; (* A287564 *)
Flatten[Position[s, 3]]; (* A287565 *)

Formula

a(n) = 4n - A287565(n) for n >= 1.