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URL: https://oeis.org/A273129

⇱ A273129 - OEIS

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A273129
The Rote-Fibonacci infinite sequence.
25
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.
Theorem 1a of Du et al. (2016) basically says that this sequence is given by taking the Fibonacci word 01001010..., changing each 0 to a 2, and then using this as the run lengths of a binary word. - Jeffrey Shallit, Aug 29 2025
LINKS
Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, and Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807[cs.FL], Nov 29 2017.
C. F. Du, H. Mousavi, E. Rowland, L. Schaeffer, and Jeffrey Shallit, Decision algorithms for Fibonacci-automatic words, II: related sequences and avoidability, preprint, February 10 2016. [This article appeared in "Theoretical Computer Science" - see the URL https://doi.org/10.1016/j.tcs.2016.10.005, which seems to be behind a firewall. - N. J. A. Sloane, Aug 29 2025]
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
KEYWORD
nonn
AUTHOR
Jeffrey Shallit, May 16 2016
STATUS
approved