A304588 - OEIS

%I #20 Nov 04 2020 11:22:28

%S 3,5,9,10,17,18,19,20,33,34,35,36,37,38,39,40,65,66,67,68,69,70,71,72,

%T 73,74,75,76,77,78,79,80,129,130,131,132,133,134,135,136,137,138,139,

%U 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

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

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

%C 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

%H Rémy Sigrist, <a href="/A304588/b304588.txt">Table of n, a(n) for n = 1..4096</a>

%H Yann Bugeaud and Dong Han Kim, <a href="https://arxiv.org/abs/1510.00282">On the b-ary expansions of log(1+1/a) and e</a>, Arxiv preprint arXiv:1510.00282 [math.NT], October 1 2015.

%H Yann Bugeaud and Dong Han Kim, <a href="https://dx.doi.org/10.2422/2036-2145.201603_002">On the b-ary expansions of log(1+1/a) and e</a>, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), 931-947.

%H Rémy Sigrist, <a href="/A304588/a304588.gp.txt">PARI program for A304588</a>

%F Apparently, a(n+1) = A004756(n) + 1. - _Rémy Sigrist_, Nov 04 2020

%e 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.

%o (PARI) See Links section.

%Y Cf. A010060, A004756.

%K nonn

%O 1,1

%A _Jeffrey Shallit_, May 15 2018