A304588 Length of shortest prefix of the Thue-Morse word (A010060) such that some length-n block appears twice.

3, 5, 9, 10, 17, 18, 19, 20, 33, 34, 35, 36, 37, 38, 39, 40, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266

Offset

1,1

Comments

lim inf a(n)/n = 5/2 and lim sup a(n)/n = 4.

a(n) is "2-sychronized", which means that there is an automaton that accepts, in parallel, the base-2 expansions of n and a(n). For this sequence an 8-state automaton suffices. - Jeffrey Shallit, Mar 06 2020

Links

Rémy Sigrist, Table of n, a(n) for n = 1..4096

Yann Bugeaud and Dong Han Kim, On the b-ary expansions of log(1+1/a) and e, Arxiv preprint arXiv:1510.00282 [math.NT], October 1 2015.

Yann Bugeaud and Dong Han Kim, On the b-ary expansions of log(1+1/a) and e, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), 931-947.

Rémy Sigrist, PARI program for A304588

Formula

Apparently, a(n+1) = A004756(n) + 1. - Rémy Sigrist, Nov 04 2020

Example

For n = 3 we have a(3) = 9 because the first 9 symbols of Thue-Morse are 011010011, and 011 is the first length-3 prefix to be repeated in this prefix.

Prog

(PARI) See Links section.

Crossrefs

Cf. A010060, A004756.

Sequence in context: A344293 A063038 A236309 * A335058 A066769 A323765

Adjacent sequences: A304585 A304586 A304587 * A304589 A304590 A304591

Keyword

nonn

Author

Jeffrey Shallit, May 15 2018

Status

approved