A216955 -id:A216955 - cp's OEIS frontend

A211967 Triangle of decimal equivalents of binary numbers with no initial repeats, A211027.

1, 2, 4, 5, 8, 9, 11, 16, 17, 18, 19, 22, 23, 32, 33, 34, 35, 37, 38, 39, 44, 46, 47, 64, 65, 66, 67, 68, 69, 70, 71, 74, 75, 76, 77, 78, 79, 88, 89, 92, 93, 94, 95, 128, 129, 130, 131, 132, 133, 134, 135, 137, 138, 139, 140, 141, 142, 143, 148, 149, 150

Offset: 1

Omar E. Pol, Nov 30 2012

			Irregular triangle begins:
1;
2;
4,   5;
8,   9, 11;
16, 17, 18, 19, 22, 23;
32, 33, 34, 35, 37, 38, 39, 44, 46, 47;

Alois P. Heinz, Rows n = 1..15, flattened
Index entries for sequences related to curling numbers

Columns 1-2 give: A000079(n-1), A000051(n-1) for n>2. Row n has length A093371(n). Right border gives A083329(n-1).

Cf. A122536, A211029, A211968, A211969, A216955.

s:= proc(n) s(n):= `if`(n=1, [[1]], map(x->
      [[x[], 0], [x[], 1]][], s(n-1))) end:
T:= proc(n) map (x-> add(x[i]*2^(nops(x)-i), i=1..nops(x)), select
      (proc(l) local i; for i to iquo(nops(l), 2) do if l[1..i]=
      l[i+1..2*i] then return false fi od; true end, s(n)))[] end:
seq (T(n), n=1..8);  # Alois P. Heinz, Dec 03 2012

A211973 a(n) = A121880(2*n)/2.

Original entry on oeis.org

1, 5, 22, 91, 369, 1486, 5962, 23884, 95607, 382568, 1530552, 6122765, 24492171, 97970902, 391888040, 1567561019, 6270261786, 25081082556, 100324401036, 401297745749, 1605191266193, 6420765631136, 25683063657239, 102732256894319

Offset: 1

Omar E. Pol, Nov 30 2012

nonn

Bisection of A093370.

Cf. A093371, A122536, A216955.

More terms from Hakan Icoz, Sep 04 2020

A217211 Number of binary sequences of length n with curling number 2.

Original entry on oeis.org

0, 2, 2, 6, 12, 26, 52, 110, 214, 438, 876, 1762, 3524, 7084, 14144, 28360, 56720, 113542, 227084, 454448, 908804, 1818168, 3636336, 7273614, 14547228, 29096678, 58192994, 116390424, 232780848, 465569860, 931139720, 1862297158, 3724592874, 7449221168, 14898442336, 29796952652, 59593905304

Offset: 1

N. J. A. Sloane, Oct 01 2012

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..104 [Entries up through 51 found by direct search (see triangle A216955), later entries obtained from a recurrence.]

Column 2 of A216955. Cf. A122536, A217212.

a(33)-a(37) from Allan Wilks, Oct 06 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.

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.

A217212 Number of sequences of 2's and 3's of length n with curling number 3.

Original entry on oeis.org

0, 0, 2, 2, 4, 10, 20, 38, 82, 164, 328, 660, 1320, 2640, 5304, 10596, 21192, 42424, 84848, 169668, 339428, 678856, 1357712, 2715548, 5431096, 10862192, 21724746, 43449380, 86898760, 173798220, 347596440, 695192670

Offset: 1

N. J. A. Sloane, Oct 01 2012

nonn

Column 3 of A216955. Cf. A122536, A217211.

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.

A211975 A122536(2n)/2.

Original entry on oeis.org

1, 3, 10, 37, 143, 562, 2230, 8884, 35465, 141720, 566600, 2265843, 9062261, 36246826, 144982872, 579922629, 2319672806, 9278655812, 37114552436, 148458068139, 593831989359, 2375327391072, 9501308431593, 38005231461009, 152020921313377

Offset: 1

Omar E. Pol, Nov 30 2012

nonn

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.
Index entries for sequences related to curling numbers

Bisection of A093371.

Cf. A093370, A121880, A216955.

cp's OEIS Frontend

A211967 Triangle of decimal equivalents of binary numbers with no initial repeats, A211027.

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A211973 a(n) = A121880(2*n)/2.

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A217211 Number of binary sequences of length n with curling number 2.

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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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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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A217212 Number of sequences of 2's and 3's of length n with curling number 3.

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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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A211975 A122536(2n)/2.

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