Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #67 Oct 31 2024 06:40:12
%S 2,2,4,6,12,20,40,74,148,286,572,1124,2248,4460,8920,17768,35536,
%T 70930,141860,283440,566880,1133200,2266400,4531686,9063372,18124522,
%U 36249044,72493652,144987304,289965744
%N Number of binary sequences of length n with no initial repeats (or, with no final repeats).
%C An initial repeat of a string S is a number k>=1 such that S(i)=S(i+k) for i=0..k-1. In other words, the first k symbols are the same as the next k symbols, e.g., ABCDABCDZQQ has an initial repeat of size 4.
%C Equivalently, this is the number of binary sequences of length n with curling number 1. See A216955. - _N. J. A. Sloane_, Sep 26 2012
%H Allan Wilks, <a href="/A122536/b122536.txt">Table of n, a(n) for n = 1..200</a> (The first 71 terms were computed by _N. J. A. Sloane_.)
%H B. Chaffin, J. P. Linderman, N. J. A. Sloane and Allan Wilks, <a href="http://arxiv.org/abs/1212.6102">On Curling Numbers of Integer Sequences</a>, arXiv:1212.6102 [math.CO], 2012-2013.
%H B. Chaffin, J. P. Linderman, N. J. A. Sloane and Allan Wilks, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Sloane/sloane3.html">On Curling Numbers of Integer Sequences</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.4.3.
%H Daniel Gabric, Jeffrey Shallit, <a href="https://arxiv.org/abs/1906.03689">Borders, Palindrome Prefixes, and Square Prefixes</a>, arXiv:1906.03689 [cs.DM], 2019.
%H Guy P. Srinivasan, <a href="/A122536/a122536.txt">Java program for this sequence and A003000</a>
%H <a href="/index/Cu#curling_numbers">Index entries for sequences related to curling numbers</a>
%F Conjecture: a_n ~ C * 2^n where C is 0.27004339525895354325... [Chaffin, Linderman, Sloane, Wilks, 2012]
%F a(2n+1)=2*a(2n) = A211965(n+1), a(2n)=2*a(2n-1)-A216958(n) = A211966(n). - _N. J. A. Sloane_, Sep 28 2012
%F a(1) = 2; a(2n) = 2*[a(2n-1) - A216959(n)], n >= 1. - _Daniel Forgues_, Feb 25 2015
%e a(4)=6: 0100, 0110, 0111, 1000, 1001 and 1011. (But not 00**, 11**, 0101, 1010.)
%Y Twice A093371. Leading column of each of the triangles A216955, A217209, A218869, A218870. Different from, but easily confused with, A003000 and A216957. - _N. J. A. Sloane_, Sep 26 2012
%Y See A121880 for difference from 2^n.
%K nonn
%O 1,1
%A _Guy P. Srinivasan_, Sep 18 2006
%E a(31)-a(71) computed from recurrence and the first 30 terms of A216958 by _N. J. A. Sloane_, Sep 28 2012, Oct 25 2012