A218869 -id:A218869 - cp's OEIS frontend

A218871 Second column of A218869.

0, 2, 4, 12, 20, 52, 100, 214, 414, 876, 1722, 3524, 6992, 14144, 28188, 56720, 113180, 227084, 453748, 908804, 1816726, 3636336, 7270770, 14547228, 29090912, 58192994, 116378984, 232780848, 465546796, 931139720, 1862251212, 3724592874, 7449128940, 14898442336, 29796768552

Offset: 2

N. J. A. Sloane, Nov 07 2012

nonn

John P. Linderman, Table of n, a(n) for n = 2..64
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.

Cf. A218869.

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

Original entry on oeis.org

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

Offset: 1

Sarah Nibs, Sep 18 2006

nonn

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

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

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.
Sarah Nibs, Java program for this sequence and A003000
Index entries for sequences related to curling numbers

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.

Conjecture:  a_n ~ C * 2^n where C is 0.27004339525895354325... [Chaffin, Linderman, Sloane, Wilks, 2012]
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
a(1) = 2; a(2n) = 2*[a(2n-1) - A216959(n)], n >= 1. - Daniel Forgues, Feb 25 2015

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

A218870 Triangle read by rows: T(n,k) = number of aperiodic binary sequences of length n with curling number <= k (1 <= k <= n).

Original entry on oeis.org

2, 2, 2, 4, 6, 6, 6, 10, 12, 12, 12, 24, 28, 30, 30, 20, 40, 48, 52, 54, 54, 40, 92, 112, 120, 124, 126, 126, 74, 174, 210, 226, 234, 238, 240, 240, 148, 362, 438, 474, 490, 498, 502, 504, 504, 286, 700, 860, 928, 960, 976, 984, 988, 990, 990, 572, 1448, 1776, 1916, 1984, 2016, 2032, 2040, 2044, 2046, 2046

Offset: 1

N. J. A. Sloane, Nov 07 2012

S is aperiodic if it is not of the form S = T^m with m > 1.
Rows are partial sums of rows of A218869.
Final entries in rows form A027375. First column is A122536.

			Triangle begins:
[2]
[2, 2]
[4, 6, 6]
[6, 10, 12, 12]
[12, 24, 28, 30, 30]
[20, 40, 48, 52, 54, 54]
[40, 92, 112, 120, 124, 126, 126]
[74, 174, 210, 226, 234, 238, 240, 240]
[148, 362, 438, 474, 490, 498, 502, 504, 504]
[286, 700, 860, 928, 960, 976, 984, 988, 990, 990]
[572, 1448, 1776, 1916, 1984, 2016, 2032, 2040, 2044, 2046, 2046]
...

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.
N. J. A. Sloane, Rows 1 through 36
Index entries for sequences related to curling numbers

Cf. A216955, A122536, A027375, A218869.

A218875 Triangle read by rows: T(n,k) (1 <= k <= n) = number of robust primitive binary sequences of length n and curling number k.

Original entry on oeis.org

2, 2, 0, 4, 2, 0, 6, 4, 2, 0, 10, 12, 4, 2, 0, 20, 20, 8, 4, 2, 0, 36, 52, 20, 8, 4, 2, 0, 72, 98, 36, 16, 8, 4, 2, 0, 142, 214, 76, 36, 16, 8, 4, 2, 0, 280, 414, 160, 68, 32, 16, 8, 4, 2, 0, 560, 870, 326, 140, 68, 32, 16, 8, 4, 2, 0, 1114, 1720, 640, 276, 132, 64, 32, 16, 8, 4, 2, 0

Offset: 1

N. J. A. Sloane, Nov 15 2012

			Triangle begins:
[2],
[2, 0],
[4, 2, 0],
[6, 4, 2, 0],
[10, 12, 4, 2, 0],
[20, 20, 8, 4, 2, 0],
[36, 52, 20, 8, 4, 2, 0],
[72, 98, 36, 16, 8, 4, 2, 0],
[142, 214, 76, 36, 16, 8, 4, 2, 0],
[280, 414, 160, 68, 32, 16, 8, 4, 2, 0],
...

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.
N. J. A. Sloane, First 36 rows of table
Index entries for sequences related to curling numbers

Cf. A216955, A218869, A218876. First column is A216958.

The triangle in A218869 is the sum of triangles A218875 and A218876.

A218876 Triangle read by rows: T(n,k) (1 <= k <= n) = number of non-robust primitive binary sequences of length n and curling number k.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 6, 2, 0, 0, 0, 0, 0, 0, 0, 0, 10, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 26, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane, Nov 15 2012

			Triangle begins:
[0],
[0, 0],
[0, 0, 0],
[0, 0, 0, 0],
[2, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0],
[4, 0, 0, 0, 0, 0, 0],
[2, 2, 0, 0, 0, 0, 0, 0],
[6, 0, 0, 0, 0, 0, 0, 0, 0],
[6, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[12, 6, 2, 0, 0, 0, 0, 0, 0, 0, 0],
[10, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
...

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.
N. J. A. Sloane, On Curling Numbers of Integer Sequences, Combinatorics on Words Conference, Fields Institute, Toronto, April 22, 2013.
N. J. A. Sloane, First 36 rows of table
Index entries for sequences related to curling numbers

Cf. A216955, A218869, A218875.

The triangle in A218869 is the sum of triangles A218875 and A218876.

cp's OEIS Frontend

A218871 Second column of A218869.

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

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A218870 Triangle read by rows: T(n,k) = number of aperiodic binary sequences of length n with curling number <= k (1 <= k <= n).

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A218875 Triangle read by rows: T(n,k) (1 <= k <= n) = number of robust primitive binary sequences of length n and curling number k.

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A218876 Triangle read by rows: T(n,k) (1 <= k <= n) = number of non-robust primitive binary sequences of length n and curling number k.

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