A093370 -id:A093370 - cp's OEIS frontend

A093371 Start with any initial string of n numbers s(1), ..., s(n), with s(1) = 2, other s(i)'s = 2 or 3 (so there are 2^(n-1) starting strings). The rule for extending the string is this as follows: To get s(n+1), write the string s(1)s(2)...s(n) as xy^k for words x and y (where y has positive length) and k is maximized, i.e., k = the maximal number of repeating blocks at the end of the sequence. Then a(n) = number of starting strings for which k = 1.

1, 1, 2, 3, 6, 10, 20, 37, 74, 143, 286, 562, 1124, 2230, 4460, 8884, 17768, 35465, 70930, 141720, 283440, 566600, 1133200, 2265843, 4531686, 9062261, 18124522, 36246826, 72493652, 144982872

Offset: 1

N. J. A. Sloane, Apr 28 2004

nonn

See A122536 for many more terms. - N. J. A. Sloane, Oct 25 2012

F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence, J. Integer Sequences, Vol. 10 (2007), #07.1.2.
F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence [pdf, ps].
Index entries for sequences related to curling numbers

Cf. A093370, A093369, A090822, A216955, A216956.
Equals A122536/2. - N. J. A. Sloane, Sep 25 2012
Different from, but easily confused with, A007148 and A045690.

a(n) = 2^(n-1) - A093370(n).

More terms from N. J. A. Sloane, Sep 26 2012

A121880 a(n) = 2^n - A122536(n).

Original entry on oeis.org

0, 2, 4, 10, 20, 44, 88, 182, 364, 738, 1476, 2972, 5944, 11924, 23848, 47768, 95536, 191214, 382428, 765136, 1530272, 3061104, 6122208, 12245530, 24491060, 48984342, 97968684, 195941804, 391883608, 783776080, 1567552160, 3135122038, 6270244076

Offset: 1

N. J. A. Sloane, Sep 21 2006

nonn

Number of binary sequences of length n with curling number > 1. See A122536 for much more information.

Robert Price, Table of n, a(n) for n = 1..200
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.
Index entries for sequences related to curling numbers

Cf. A122536. Similar to but different from A094536.

A093369 a(n) = sum of lengths of strings that can be generated by any starting string of n 2's and 3's that starts with a 2, using the rule described in the Comments lines.

Original entry on oeis.org

1, 6, 14, 42, 98, 242, 552, 1394, 2935, 6471, 14006, 30060, 64223, 136914, 290224, 613509, 1292567, 2717311, 5696864, 11920124, 24889066, 51880008, 107954163, 224305440, 465388743, 964349526, 1995808823, 4125871527, 8520180124, 17577302639, 36228352911

Offset: 1

N. J. A. Sloane, Apr 28 2004

nonn

Start with any initial string of n numbers s(1), ..., s(n), with s(1) = 2, other s(i)'s = 2 or 3 (so there are 2^(n-1) starting strings). The rule for extending the string is this:
To get s(i+1), write the string s(1)s(2)...s(i) as xy^k for words x and y (where y has positive length) and k is maximized, i.e. k = the maximal number of repeating blocks at the end of the sequence so far. Then s(i+1) = k if k >=2, but if k=1 you must stop (without writing down the 1).
a(n) = sum of final length of string, summed over all 2^(n-1) starting strings.

			a(3) = 14: the starting string, final string and length are as follows:
222 2223 4
223 223 3
232 232 3
233 2332 4, for a total of 4+3+3+4 = 14.

Lars Blomberg, Table of n, a(n) for n = 1..37
F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence, J. Integer Sequences, Vol. 10 (2007), #07.1.2.
F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence [pdf, ps].
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 Gijswijt's sequence
Index entries for sequences related to curling numbers

Cf. A090822, A093370, A093371, A094004, A094005.

a(21)-a(31) from Lars Blomberg, Jul 25 2017

A094005 a(n) = sum of lengths of strings that can be generated by any starting string of n 2's and 3's, using the rule described in the Comments lines.

Original entry on oeis.org

2, 11, 30, 82, 199, 480, 1097, 2630, 5828, 12830, 27873, 60071, 128355, 273543, 580149, 1226626, 2584822, 5433676, 11392986, 23838396, 49776503, 103755527, 215904926, 448602871, 930771041, 1928682932, 3991605129, 8251710234, 17040335019, 35154540729, 72456654860, 149208536983

Offset: 1

N. J. A. Sloane, May 31 2004

nonn

Start with any initial string of n numbers s(1), ..., s(n), all = 2 or 3 (so there are 2^n starting strings). The rule for extending the string is this:
To get s(i+1), write the string s(1)s(2)...s(i) as xy^k for words x and y (where y has positive length) and k is maximized, i.e., k = the maximal number of repeating blocks at the end of the sequence so far (k is the curling number of s(1)s(2)...s(i)). Then s(i+1) = k if k >=2, but if k=1 you must stop (without writing down the 1).
a(n) = sum of final length of string, summed over all 2^n starting strings.
See A094004 for more terms. - N. J. A. Sloane, Dec 25 2012

F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence, J. Integer Sequences, Vol. 10 (2007), #07.1.2.
F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence [pdf, ps].
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 Gijswijt's sequence
Index entries for sequences related to curling numbers

Cf. A090822, A093370, A093371, A094004, A216813.

Equals A216813(n) + n*2^n. - N. J. A. Sloane, Sep 26 2012

A093369 is closely related.

a(27)-a(31) from N. J. A. Sloane, Sep 19 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

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

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

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A121880 a(n) = 2^n - A122536(n).

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A093369 a(n) = sum of lengths of strings that can be generated by any starting string of n 2's and 3's that starts with a 2, using the rule described in the Comments lines.

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A094005 a(n) = sum of lengths of strings that can be generated by any starting string of n 2's and 3's, using the rule described in the Comments lines.

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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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Programs

Maple

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

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A211975 A122536(2n)/2.

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