A218870 -id:A218870

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A122536

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

+10
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.

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(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.

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

A218869

Triangle read by rows: T(n,k) = number of aperiodic binary sequences of length n with curling number k (1 <= k <= n).

+10
6

2, 2, 0, 4, 2, 0, 6, 4, 2, 0, 12, 12, 4, 2, 0, 20, 20, 8, 4, 2, 0, 40, 52, 20, 8, 4, 2, 0, 74, 100, 36, 16, 8, 4, 2, 0, 148, 214, 76, 36, 16, 8, 4, 2, 0, 286, 414, 160, 68, 32, 16, 8, 4, 2, 0, 572, 876, 328, 140, 68, 32, 16, 8, 4, 2, 0, 1124, 1722, 640, 276, 132, 64, 32, 16, 8, 4, 2, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

S is aperiodic if it is not of the form S = T^m with m > 1.

Row sums are A027375. First column is A122536.

It appears that reversed rows converge to A155559. - Omar E. Pol, Nov 20 2012

LINKS

Table of n, a(n) for n=1..78.

B. Chaffin, J. P. Linderman, N. J. A. Sloane and Allan Wilks, On Curling Numbers of Integer Sequences, arXiv:1212.6102, Dec 25 2012.

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.

John P. Linderman, Rows 1 through 64 (Rows 1 through 36 were computed by N. J. A. Sloane)

Index entries for sequences related to curling numbers

EXAMPLE

Triangle begins:

2, 0,

4, 2, 0,

6, 4, 2, 0,

12, 12, 4, 2, 0,

20, 20, 8, 4, 2, 0,

40, 52, 20, 8, 4, 2, 0,

74, 100, 36, 16, 8, 4, 2, 0,

148, 214, 76, 36, 16, 8, 4, 2, 0,

286, 414, 160, 68, 32, 16, 8, 4, 2, 0,

572, 876, 328, 140, 68, 32, 16, 8, 4, 2, 0,

...

CROSSREFS

Cf. A216955, A122536, A027375, A218870.

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Nov 07 2012

STATUS

approved

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