A273129 The Rote-Fibonacci infinite sequence.

0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1

Offset

0

Comments

This is an aperiodic sequence that avoids the pattern x x x^R, where x is a nonempty block and x^R denotes the reversal of x.

It can be generated as the limit of the words R(i), where R(0) = 0, R(1) = 00, and R(n) = R(n-1)R(n-2) if n == 0 (mod 3), and R(n) = R(n-1) c(R(n-2)) if n == 1, 2 (mod 3), where c flips 0 to 1 and vice versa.

It can also be generated as the image, under the coding that maps a, b -> 0 and c, d -> 1, of the fixed point (see A316340), starting with a, of the morphism a -> abcab, b -> cda, c -> cdacd, d -> abc.

Links

Paolo Xausa, Table of n, a(n) for n = 0..6764

Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807, Nov 29 2017

C. F. Du, H. Mousavi, E. Rowland, L. Schaeffer, J. Shallit, Decision algorithms for Fibonacci-automatic words, II: related sequences and avoidability, preprint, February 10 2016.

Mathematica

Quotient[First[SubstitutionSystem[{0 -> {0, 1, 2, 0, 1}, 1 -> {2, 3, 0}, 2 -> {2, 3, 0, 2, 3}, 3 -> {0, 1, 2}}, {0}, {4}]], 2] (* Paolo Xausa, Jan 30 2025 *)

Crossrefs

Cf. A316340.

Sequence in context: A189640 A289057 A106138 * A288936 A064990 A284388

Adjacent sequences: A273126 A273127 A273128 * A273130 A273131 A273132

Keyword

nonn,changed

Author

Jeffrey Shallit, May 16 2016

Status

approved