

A122536


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


24



2, 2, 4, 6, 12, 20, 40, 74, 148, 286, 572, 1124, 2248, 4460, 8920, 17768, 35536, 70930, 141860, 283440, 566880, 1133200, 2266400, 4531686, 9063372, 18124522, 36249044, 72493652, 144987304, 289965744
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OFFSET

1,1


COMMENTS

An initial repeat of a string S is a number k>=1 such that S(i)=S(i+k) for i=0..k1. 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.
Equivalently, this is the number of binary sequences of length n with curling number 1. See A216955.  N. J. A. Sloane, Sep 26 2012


LINKS

Allan Wilks, Table of n, a(n) for n = 1..200 (The first 71 terms were computed by N. J. A. Sloane.)
B. Chaffin, J. P. Linderman, N. J. A. Sloane and Allan Wilks, On Curling Numbers of Integer Sequences, arXiv:1212.6102 [math.CO], 20122013.
B. Chaffin, J. P. Linderman, N. J. A. Sloane and Allan Wilks, On Curling Numbers of Integer Sequences, Journal of Integer Sequences, Vol. 16 (2013), Article 13.4.3.
Daniel Gabric, Jeffrey Shallit, Borders, Palindrome Prefixes, and Square Prefixes, arXiv:1906.03689 [cs.DM], 2019.
Guy P. Srinivasan, Java program for this sequence and A003000
Index entries for sequences related to curling numbers


FORMULA

Conjecture: a_n ~ C * 2^n where C is 0.27004339525895354325... [Chaffin, Linderman, Sloane, Wilks, 2012]
a(2n+1)=2*a(2n), a(2n)=2*a(2n1)A216958(n).  N. J. A. Sloane, Sep 28 2012
a(1) = 2; a(2n) = 2*[a(2n1)  A216959(n)], a(2n+1) = 2*a(2n), n >= 1.  Daniel Forgues, Feb 25 2015


EXAMPLE

a(4)=6: 0100, 0110, 0111, 1000, 1001 and 1011. (But not 00**, 11**, 0101, 1010.)


CROSSREFS

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
See A121880 for difference from 2^n.
Sequence in context: A063886 A003000 A216957 * A238014 A052953 A128209
Adjacent sequences: A122533 A122534 A122535 * A122537 A122538 A122539


KEYWORD

nonn


AUTHOR

Guy P. Srinivasan, Sep 18 2006


EXTENSIONS

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


STATUS

approved



