proposed

approved

editing

proposed

allocated for Clark KimberlingStart with 0 and repeatedly substitute 0->0321, 1->3210, 2->2103, 3->1032.

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

1,2

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.

Clark Kimberling, <a href="/A287576/b287576.txt">Table of n, a(n) for n = 1..20000</a>

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

a(n) = 4n - A287578(n) for n >= 1.

First three iterations of the morphism:

0321

0321103221033210

0321103221033210321003211032210321033210032110321032210332100321

s = Nest[Flatten[# /. {0 -> {0, 3, 2, 1}, 1 -> {3, 2, 1, 0}, 2 -> {2, 1, 0, 3}, 3 -> {1, 0, 3, 2}}] &, {0}, 9]; (* A287576 *)

Flatten[Position[s, 0]]; (* A287577 *)

Flatten[Position[s, 1]]; (* A287578 *)

Flatten[Position[s, 2]]; (* A287579 *)

Flatten[Position[s, 3]]; (* A287580 *)

Cf. A287577, A287578, A287579, A287580.

allocated

nonn,easy

Clark Kimberling, Jun 01 2017

approved

editing

allocated for Clark Kimberling

allocated

approved