%I #69 Dec 12 2022 08:18:50
%S 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,
%T 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,
%U 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
%N 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.
%C 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.
%C Cousin of the so-called 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.
%C This has some similarities with Gijswijt's sequence A090822. - _N. J. A. Sloane_, Dec 21 2020
%H Rémy Sigrist, <a href="/A337546/b337546.txt">Table of n, a(n) for n = 1..10000</a>
%H Alessandro Della Corte, <a href="/A337546/a337546_1.txt">Octave program</a>
%H Alessandro Della Corte, <a href="https://www.researchgate.net/publication/361085516_The_Easily_Bored_Sequence">The Easily Bored Sequence</a>, Univ. of Camerino (2022).
%H Alessandro Della Corte, <a href="https://doi.org/10.1016/j.topol.2022.108244">The Easily Bored Sequence</a>, Topology and Its Applications 320 (2022), 108244.
%H MathOverflow, <a href="https://mathoverflow.net/questions/377105/the-easily-bored-sequence">The easily bored sequence</a>
%H Rémy Sigrist, <a href="/A337546/a337546_2.txt">C program for A337546</a>
%e 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.
%o (C) See Links section.
%Y Cf. A006345, A090822.
%K nonn
%O 1
%A _Alessandro Della Corte_, Nov 22 2020