A363057 Run lengths of the Fibonacci word (A003849).

1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2

Offset

1,3

Comments

The sequence is generated by applying the coding 0->1 1->1 2->2 3->2 4->1 5->1 6->1 7->2 8->1 9->1 to the fixed point of the sequence generated by iterating the morphism 0->01 1->2 2->45 3->91 4->67 5->4 6->28 7->9 8->6 9->31. Alternatively, there is a 10-state automaton to compute the n-th term (where the input is the Zeckendorf representation of n).

Links

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Formula

a(2*n) = 1. - A.H.M. Smeets, Mar 31 2024

Example

The first 6 terms of A003849 are 0,1,0,0,1,0 so the first 4 terms of the run-length encoding are 1,1,2,1.

Mathematica

Map[Length, Most[Split[Nest[Flatten[ReplaceAll[#, {0 -> {0, 1}, 1 -> 0}]] &, 0, 10]]]] (* Paolo Xausa, Apr 30 2024 *)

Crossrefs

Cf. A003849, A096270.

Sequence in context: A263025 A184348 A307614 * A242481 A228287 A213636

Adjacent sequences: A363054 A363055 A363056 * A363058 A363059 A363060

Keyword

nonn

Author

Jeffrey Shallit, May 15 2023

Status

approved