A079101 A repetition-resistant sequence.

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

Offset

1,1

Comments

a(n) = 0 or 1, chosen so as to maximize the number of different subsequences that are formed.

a(n+1)=1 if and only if (a(1),a(2),...,a(n),0), but not (a(1),a(2),...,a(n),1), has greater length of longest repeated segment than (a(1),a(2),...,a(n)) has.

In Feb, 2003, Alejandro Dau solved Problem 3 on the Unsolved Problems and Rewards website, thus establishing that every binary word occurs infinitely many times in this sequence.

Klaus Sutmer remarks (Jun 26 2006) that this sequence is very similar to the Ehrenfeucht-Mycielski sequence A007061. Both sequences have every finite binary word as a factor; in fact, essentially the same proof works for both sequences.

Differs from A334941 for the first time at n = 70. - Jeffrey Shallit, Dec 14 2022

Links

Peter J. C. Moses, Table of n, a(n) for n = 1..10000

A. Dau, Secuencia Maximizadora de Subcadenas (Interactive Java generator of repetition-resistant sequences). [Broken link]

Clark Kimberling, Unsolved Problems and Rewards.

Clark Kimberling, Problem 2289, Crux Mathematicorum 23 (1997) 501.

Example

a(7)=1 because (0,1,0,0,0,1,0) has repeated segment (0,1,0) of length 3, whereas (0,1,0,0,0,1,1) has no repeated segment of length 3.

Crossrefs

Cf. A079136, A079335, A079336, A079337, A079338, A007061, A334941.

Sequence in context: A341258 A285831 A188294 * A334941 A076478 A091444

Adjacent sequences: A079098 A079099 A079100 * A079102 A079103 A079104

Keyword

nonn

Author

Clark Kimberling, Jan 03 2003

Status

approved