A122536 -id:A122536 - cp's OEIS frontend

A218869 Triangle read by rows: T(n,k) = number of aperiodic binary sequences of length n with curling number k (1 <= k <= n).

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

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.
Row sums are A027375. First column is A122536.
It appears that reversed rows converge to A155559. - Omar E. Pol, Nov 20 2012

			Triangle begins:
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,
...

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

Cf. A216955, A122536, A027375, A218870.

A211029 Triangle read by rows in which row n lists the binary words of length n over the alphabet {1,2} with no initial repeats.

Original entry on oeis.org

1, 2, 12, 21, 121, 122, 211, 212, 1211, 1221, 1222, 2111, 2112, 2122, 12111, 12112, 12211, 12212, 12221, 12222, 21111, 21112, 21121, 21122, 21221, 21222, 121111, 121112, 121122, 122111, 122112, 122121, 122211, 122212, 122221, 122222, 211111, 211112

Offset: 1

Omar E. Pol, Nov 29 2012

As usual in the OEIS, binary alphabets are encoded with {1,2} over the alphabet {0,1} the entries contain nonzero "numbers" beginning with 0.

			The fourth row of triangle of binary sequences is
0100, 0110, 0111, 1000, 1001, 1011 (see section example of A122536) therefore the fourth row of this triangle is
1211, 1221, 1222, 2111, 2112, 2122.
The first six rows of triangle are:
1, 2;
12, 21;
121, 122, 211, 212;
1211, 1221, 1222, 2111, 2112, 2122;
12111, 12112, 12211, 12212, 12221, 12222, 21111, 21112, 21121, 21122, 21221, 21222;
121111, 121112, 121122, 122111, 122112, 122121, 122211, 122212, 122221, 122222, 211111, 211112, 211121, 211122, 211212, 211221, 211222, 212211, 212221, 212222;

Alois P. Heinz, Rows n = 1..14, flattened
Index entries for sequences related to curling numbers

Row n has length A122536(n).

Cf. A093371, A211027, A213969, A216955.

s:= proc(n) s(n):= `if`(n=0, [[]], map(x->
      [[x[], 1], [x[], 2]][], s(n-1))) end:
T:= proc(n) map(x-> parse(cat(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..7);  # Alois P. Heinz, Dec 02 2012

More terms and name improved by R. J. Mathar, Nov 30 2012

A211968 Triangle of binary numbers with some initial repeats.

Original entry on oeis.org

11, 110, 111, 1010, 1100, 1101, 1110, 1111, 10100, 10101, 11000, 11001, 11010, 11011, 11100, 11101, 11110, 11111, 100100, 101000, 101001, 101010, 101011, 101101, 110000, 110001, 110010, 110011, 110100, 110101, 110110, 110111, 111000, 111001, 111010, 111011

Offset: 2

Omar E. Pol, Dec 03 2012

Triangle read by rows in which row n lists the binary numbers with n digits and with some initial repeats, n >= 2.

Also triangle read by rows in which row n lists the binary words of length n with some initial repeats and with initial digit 1, n >= 2.

			Triangle begins, starting at row 2:
  11;
  110, 111;
  1010, 1100, 1101, 1110, 1111;
  10100, 10101, 11000, 11001, 11010, 11011, 11100, 11101, 11110, 11111;

Alois P. Heinz, Rows n = 2..14, flattened
Index entries for sequences related to curling numbers

Complement in base 2 of A211027.
Rows lengths give: A093370.
Cf. A093370, A093371, A121880, A122536, A211967, A211969, A211973, 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-> parse(cat(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 true fi od; false end, s(n)))[] end:
seq(T(n), n=2..7);  # Alois P. Heinz, Dec 04 2012

T[n_] := FromDigits /@ Select[Range[2^(n-1), 2^n-1] // IntegerDigits[#, 2]&, FindTransientRepeat[Reverse[#], 2][[2]] != {}&];
Table[T[n], {n, 2, 7}] // Flatten (* Jean-François Alcover, Feb 12 2025 *)

A211969 Triangle of decimal equivalents of binary numbers with some initial repeats, A211968.

Original entry on oeis.org

3, 6, 7, 10, 12, 13, 14, 15, 20, 21, 24, 25, 26, 27, 28, 29, 30, 31, 36, 40, 41, 42, 43, 45, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 72, 73, 80, 81, 82, 83, 84, 85, 86, 87, 90, 91, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106

Offset: 2

Omar E. Pol, Dec 03 2012

			Irregular triangle begins, starting at row 2:
3;
6, 7;
10, 12, 13, 14, 15;
20, 21, 24, 25, 26, 27, 28, 29, 30, 31;
36, 40, 41, 42, 43, 45, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63;

Alois P. Heinz, Rows n = 2..14, flattened
Index entries for sequences related to curling numbers

Complement of A211967.
Row lengths give: A093370.
Column 1 gives: A005418(n+1).
Right border gives: A000225(n).
Cf. A093370, A093371, A121880, A122536, A211027, A211029, A211973, 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 true fi od; false end, s(n)))[] end:
seq (T(n), n=2..7);  # Alois P. Heinz, Dec 04 2012

A217209 Irregular triangle read by rows: T(n,k) (n>=1, 1 <= k <= A217208(n)) = number of strings of n 2's and 3's having a tail of length k.

Original entry on oeis.org

2, 2, 1, 1, 4, 2, 2, 6, 5, 3, 1, 1, 12, 9, 6, 2, 3, 20, 18, 12, 6, 7, 0, 0, 0, 1, 40, 34, 25, 11, 14, 1, 0, 1, 2, 74, 71, 47, 24, 28, 1, 3, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1

Offset: 1

N. J. A. Sloane, Oct 01 2012

See A217208 or A216730 for definition of tail.

			Rows 1 through 8 are:
2,
2, 1, 1,
4, 2, 2,
6, 5, 3, 1, 1,
12, 9, 6, 2, 3,
20, 18, 12, 6, 7, 0, 0, 0, 1,
40, 34, 25, 11, 14, 1, 0, 1, 2,
74, 71, 47, 24, 28, 1, 3, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1,
148, 139, 95, 48, 56, 6, 4, 3, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 1, 1,
...

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.
Ben Chaffin and N. J. A. Sloane, Rows 1 through 48 [The first 35 rows were computed by _N. J. A. Sloane_]
Index entries for sequences related to curling numbers

Cf. A217208 (row lengths), A216813 (means), A122536 (first column), A217210 (second column), A216730, A094004, A090822.

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

Original entry on oeis.org

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.

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.

cp's OEIS Frontend

A218869 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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A211029 Triangle read by rows in which row n lists the binary words of length n over the alphabet {1,2} with no initial repeats.

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A211968 Triangle of binary numbers with some initial repeats.

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A211969 Triangle of decimal equivalents of binary numbers with some initial repeats, A211968.

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A217209 Irregular triangle read by rows: T(n,k) (n>=1, 1 <= k <= A217208(n)) = number of strings of n 2's and 3's having a tail of length k.

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

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Maple

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