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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 (list; graph; refs; listen; history; text; internal format)
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..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.
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], 2012-2013.
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.
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(2n-1)-A216958(n). - N. J. A. Sloane, Sep 28 2012
a(1) = 2; a(2n) = 2*[a(2n-1) - 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
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

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Last modified March 19 06:32 EDT 2024. Contains 370953 sequences. (Running on oeis4.)