

A337546


A binary sequence defined as follows: a(1)=0; thereafter choose a(n) so as to minimize the number of final consecutive repeated words, and if there is a choice, minimize the length of the repeated word.


3



0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1
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OFFSET

1


COMMENTS

Start with a(1)=0, then select a(n)=0 or a(n)=1 so as to minimize the largest integer T such that a(0),...,a(n) has a word repeated T times at the end, and if 0 and 1 produce the same T, so as to minimize the length of the longest word repeated T times at the end.
Cousin of the socalled Linus sequence (A006345), which avoids the longest repeated suffix. Conjectures: 1. This is cubefree (the Linus sequence is not cubefree); 2. density of 0's exists and equals 1/2; 3. recurrent and mirror invariant.
This has some similarities with Gijswijt's sequence A090822.  N. J. A. Sloane, Dec 21 2020


LINKS

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Alessandro Della Corte, Octave program
Alessandro Della Corte, The Easily Bored Sequence, Univ. of Camerino (2022).
MathOverflow, The easily bored sequence
Rémy Sigrist, C program for A337546


EXAMPLE

Consider the first case where the sequence differs from the Linus sequence, that is at n=20. Up to n=19 we have: 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0. Then inserting 0 produces a triple at the end, while inserting 1 produces two occurrences of the word 001 at the end and no cubes. Since the number of repetitions counts more than length of the repeated suffix, a(20)=1.


PROG

(C) See Links section.


CROSSREFS

Cf. A006345, A090822.
Sequence in context: A080846 A082401 A157238 * A059448 A283318 A288633
Adjacent sequences: A337543 A337544 A337545 * A337547 A337548 A337549


KEYWORD

nonn


AUTHOR

Alessandro Della Corte, Nov 22 2020


STATUS

approved



