A216958 -id:A216958 - cp's OEIS frontend

A216959 a(n) = A216958(n)/2.

1, 1, 2, 3, 5, 10, 18, 36, 71, 140, 280, 557, 1111, 2218, 4432, 8859, 17710, 35412, 70812, 141605, 283197, 566364, 1132695, 2265363, 4530659, 9061259, 18122454, 36244783, 72489435, 144978745, 289957235, 579914215, 1159828166, 2319655810, 4639311084, 9278621655

Offset: 1

N. J. A. Sloane, Sep 27 2012

nonn

Amiram Eldar, Table of n, a(n) for n = 1..100 (calculated using the b-file at A216958)
Index entries for sequences related to curling numbers.

Cf. A216958.

A216960 a(n) = A122536(n) - A216958(n).

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 4, 2, 6, 6, 12, 10, 26, 24, 56, 50, 116, 106, 236, 230, 486, 472, 1010, 960, 2054, 2004, 4136, 4086, 8434, 8254, 17018, 16828, 34184, 33992, 69056, 68314, 138754, 138020, 278174, 277452, 559152, 556246, 1121116, 1118170, 2245076, 2242058, 4501566, 4489798

Offset: 1

N. J. A. Sloane, Sep 27 2012

nonn

Cf. A122536, A216958, A216961.

a(31)-a(35) from N. J. A. Sloane, Oct 25 2012

More terms from Hakan Icoz, Dec 28 2021

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

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.

A211965 Number of binary sequences of length 2n-1 and curling number 1.

Original entry on oeis.org

2, 4, 12, 40, 148, 572, 2248, 8920, 35536, 141860, 566880, 2266400, 9063372, 36249044, 144987304, 579931488, 2319690516, 9278691224, 37114623248, 148458209744, 593832272556, 2375327957436, 9501309564288, 38005233726372, 152020925844036

Offset: 1

Omar E. Pol, Nov 28 2012

nonn

Equivalently, number of binary sequences of length 2n-1 with no initial repeats (see A122536).

B. Chaffin, J. P. Linderman, N. J. A. Sloane and Allan Wilks, On Curling Numbers of Integer Sequences, arXiv:1212.6102
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.
Index entries for sequences related to curling numbers

Bisection of A122536.

Cf. A093371, A211966, A216955, A216958.

a(n) = 2*A093371(2n-1).

a(n) = 2*A211966(n-1), n >= 2.

A211966 Number of binary sequences of length 2n and curling number 1.

Original entry on oeis.org

2, 6, 20, 74, 286, 1124, 4460, 17768, 70930, 283440, 1133200, 4531686, 18124522, 72493652, 289965744, 1159845258, 4639345612, 18557311624, 74229104872, 296916136278, 1187663978718, 4750654782144, 19002616863186, 76010462922018

Offset: 1

Omar E. Pol, Nov 28 2012

nonn

Equivalently, number of binary sequences of length 2n with no initial repeats (see A122536).

B. Chaffin, J. P. Linderman, N. J. A. Sloane and Allan Wilks, On Curling Numbers of Integer Sequences, arXiv:1212.6102
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.
Index entries for sequences related to curling numbers

Bisection of A122536.

Cf. A093371, A211965, A216955, A216958.

a(n) = 2*A093371(2n) = A093371(2n+1) = A211965(n+1)/2.

A217943 Triangle read by rows: T(n,k) = 2*C(n-1,k)-C(n,k) for kA216955(n,k).

Original entry on oeis.org

2, -2, 0, 2, -2, 2, -2, 2, -2, 0, 0, 0, 2, -2, 4, -2, -2, 0, 2, -2, 0, 0, 0, 0, 0, 2, -2, 6, -6, 2, -2, 0, 0, 2, -2, 0, 6, -6, 0, 0, 0, 0, 2, -2, 10, -10, 0, 2, -2, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, 20, -10, -4, -6, 2, -2, 0, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2

Offset: 2

N. J. A. Sloane, following a suggestion from Allan Wilks, Oct 25 2012

			Triangle begins:
[2, -2]
[0, 2, -2]
[2, -2, 2, -2]
[0, 0, 0, 2, -2]
[4, -2, -2, 0, 2, -2]
[0, 0, 0, 0, 0, 2, -2]
[6, -6, 2, -2, 0, 0, 2, -2]
[0, 6, -6, 0, 0, 0, 0, 2, -2]
[10, -10, 0, 2, -2, 0, 0, 0, 2, -2]
[0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2]
[20, -10, -4, -6, 2, -2, 0, 0, 0, 0, 2, -2]
...

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 2 through 48 of A217943
Index entries for sequences related to curling numbers

Cf. A216955.

The nonzero entries in the first column form A216958.

cp's OEIS Frontend

A216959 a(n) = A216958(n)/2.

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A216960 a(n) = A122536(n) - A216958(n).

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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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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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A211965 Number of binary sequences of length 2n-1 and curling number 1.

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A211966 Number of binary sequences of length 2n and curling number 1.

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A217943 Triangle read by rows: T(n,k) = 2*C(n-1,k)-C(n,k) for kA216955(n,k).

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