A287556 Start with 0 and repeatedly substitute 0->0132, 1->1320, 2->3201, 3->2013.

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

Offset: 1

Author

Clark Kimberling, May 31 2017

Keywords

• nonn
• easy

Comments

This is the fixed point of the morphism 0->0132, 1->1320, 2->3201, 3->2013 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.

In the following guide to related sequences, column 1 indexes fixed points on {0,1,2,3}, and column 2 indicates position sequences of 0, 1, 2, 3. Those sequences therefore comprise a 4-way splitting of the positive integers.

Fixed points of morphisms: Position sequences:

A053839: 0->0123, 1->1230, 2->2301, 3->3012 A287552-A287555

A287556: 0->0132, 1->1320, 2->3201, 3->2013 A287557-A287560

A287561: 0->0213, 1->2130, 2->1302, 3->3021 A287562-A287565

A287566: 0->0231, 1->2310, 2->3102, 3->1023 A287567-A287570

A287571: 0->0312, 1->3120, 2->1203, 3->2031 A287572-A287575

A287576: 0->0321, 1->3210, 2->2103, 3->1032 A287577-A287580

Examples

			First three iterations of the morphism:
0132
0132132020133201
0132132020133201132020133201013232010132132020132013320101321320
		

Links

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

Crossrefs

Cf. A287557, A287558, A287559, A287560.

Programs

Mathematica

s = Nest[Flatten[# /. {0 -> {0, 1, 3, 2}, 1 -> {1, 3, 2, 0}, 2 -> {3, 2, 0, 1}, 3 -> {2, 0, 1, 3}}] &, {0}, 9]; (* A287556 *)
Flatten[Position[s, 0]]; (* A287557 *)
Flatten[Position[s, 1]]; (* A287558 *)
Flatten[Position[s, 2]]; (* A287559 *)
Flatten[Position[s, 3]]; (* A287560 *)