cp's OEIS Frontend

A122536 Number of binary sequences of length n with no initial repeats (or, with no final repeats).

%I A122536 #68 Jun 14 2025 17:15:23
%S A122536 2,2,4,6,12,20,40,74,148,286,572,1124,2248,4460,8920,17768,35536,
%T A122536 70930,141860,283440,566880,1133200,2266400,4531686,9063372,18124522,
%U A122536 36249044,72493652,144987304,289965744
%N A122536 Number of binary sequences of length n with no initial repeats (or, with no final repeats).
%C A122536 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 A122536 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 A122536 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 A122536 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 A122536 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 A122536 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 A122536 Sarah Nibs, <a href="/A122536/a122536.txt">Java program for this sequence and A003000</a>
%H A122536 <a href="/index/Cu#curling_numbers">Index entries for sequences related to curling numbers</a>
%F A122536 Conjecture:  a_n ~ C * 2^n where C is 0.27004339525895354325... [Chaffin, Linderman, Sloane, Wilks, 2012]
%F A122536 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 A122536 a(1) = 2; a(2n) = 2*[a(2n-1) - A216959(n)], n >= 1. - _Daniel Forgues_, Feb 25 2015
%e A122536 a(4)=6: 0100, 0110, 0111, 1000, 1001 and 1011. (But not 00**, 11**, 0101, 1010.)
%Y A122536 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 A122536 See A121880 for difference from 2^n.
%K A122536 nonn
%O A122536 1,1
%A A122536 _Sarah Nibs_, Sep 18 2006
%E A122536 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