A216813 -id:A216813 - cp's OEIS frontend

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.

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

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.

A217208 a(n) = (conjectured) length of longest tail that can be generated by a starting string of 2's and 3's of length n before a 1 is reached, using the rule described in the Comments lines.

Original entry on oeis.org

0, 2, 2, 4, 4, 8, 8, 58, 59, 60, 112, 112, 112, 118, 118, 118, 118, 118, 119, 119, 119, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 131, 131, 131, 131, 131, 131, 131, 131, 131, 131, 131, 131, 131, 131, 131, 131, 131, 131, 131, 131, 132, 132, 132, 132, 132, 132, 132, 132, 133, 173, 173, 173, 173

Offset: 1

N. J. A. Sloane, Sep 29 2012; revised Oct 02 2012

Start with an initial string S of n numbers s(1), ..., s(n), all = 2 or 3. The rule for extending the string is this:
To get s(i+1), write the current 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 the string). Then set s(i+1) = k.
The "tail length" t(S) of S is defined as follows: start with S and repeatedly append the curling number (recomputing it at each step) until a 1 is reached; t(S) is the number of terms that are appended to S before a 1 is reached. If a 1 is never reached, set t(S)=oo .
The "Curling Number Conjecture" is that if one starts with any finite string and repeatedly extends it by appending the curling number k, then eventually one must reach a 1. This has not yet been proved.
The values shown for n >= 49 are only conjectures, because certain assumptions used to cut down the search have not yet been rigorously justified. However, we believe that ALL terms shown are correct.

			a(3) = 2, using the starting string 3,2,2, which extends to 3,2,2,2,3, of length 5.
a(4) = 4, using the starting string 2,3,2,3, which extends to 2,3,2,3,2,2,2,3 of length 8.
a(8) = 58: start = 23222323, end = 232223232223222322322232223232223222322322232223232223222322322332.
a(22) = 120: start = 2322322323222323223223: see A116909 for trajectory.

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. [pdf, ps]
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. arXiv:1212.6102
Benjamin Chaffin and N. J. A. Sloane, The Curling Number Conjecture, preprint.
Index entries for sequences related to curling numbers

a(n) = length of n-th row of A217209.
a(n) = A094004(n) - n.
Cf. A091787, A090822, A093369, A094005, A116909, A160766, A216730, A216813.

cp's OEIS Frontend

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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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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A217208 a(n) = (conjectured) length of longest tail that can be generated by a starting string of 2's and 3's of length n before a 1 is reached, using the rule described in the Comments lines.

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Crossrefs