%I #37 Aug 05 2018 11:38:41

%S 1,4,5,8,9,14,15,66,68,70,123,124,125,132,133,134,135,136,138,139,140,

%T 142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,

%U 159,160,161,162,163,164,165,166,167,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,200,201,202,203,204,205,206,207,209,250,251,252,253

%N a(n) = (conjectured) length of longest string that can be generated by a starting string of 2's and 3's of length n, using the rule described in the Comments lines.

%C Start with an initial string of n numbers s(1), ..., s(n), all = 2 or 3. The rule for extending the string is this:

%C 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 the string). Then set s(i+1) = k if k >=2, but if k=1 you must stop (without writing down the 1).

%C 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.

%C 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. - _N. J. A. Sloane_, Sep 17 2012

%H F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Sloane/sloane55.html">A Slow-Growing Sequence Defined by an Unusual Recurrence</a>, J. Integer Sequences, Vol. 10 (2007), #07.1.2.

%H 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 [<a href="http://neilsloane.com/doc/gijs.pdf">pdf</a>, <a href="http://neilsloane.com/doc/gijs.ps">ps</a>].

%H B. Chaffin, J. P. Linderman, N. J. A. Sloane and Allan Wilks, <a href="http://arxiv.org/abs/1212.6102">On Curling Numbers of Integer Sequences</a>, arXiv:1212.6102 [math.CO], Dec 25 2012.

%H B. Chaffin, J. P. Linderman, N. J. A. Sloane and Allan Wilks, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Sloane/sloane3.html">On Curling Numbers of Integer Sequences</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.4.3.

%H Benjamin Chaffin and N. J. A. Sloane, <a href="http://neilsloane.com/doc/CNC.pdf">The Curling Number Conjecture</a>, preprint.

%H <a href="/index/Cu#curling_numbers">Index entries for sequences related to curling numbers</a>

%e a(3) = 5, using the starting string 3,2,2, which extends to 3,2,2,2,3, of length 5.

%e a(4) = 8, using the starting string 2,3,2,3, which extends to 2,3,2,3,2,2,2,3 of length 8.

%e a(8) = 66: start = 23222323, end = 232223232223222322322232223232223222322322232223232223222322322332.

%e a(22) = 142: start = 2322322323222323223223: see A116909 for trajectory.

%Y Cf. A091787, A090822, A093369, A094005, A116909, A160766, A216730, A217208.

%K nonn,nice,hard

%O 1,2

%A _N. J. A. Sloane_, May 31 2004. Revised by _N. J. A. Sloane_, Sep 17 2012

%E a(27)-a(30) from _Allan Wilks_, Jul 29 2004

%E a(31)-a(36) from _Benjamin Chaffin_, Apr 09 2008

%E a(37)-a(44) (computed in 2008) from _Benjamin Chaffin_, Dec 04 2009

%E a(45)-a(48) from _Benjamin Chaffin_, Dec 18 2009

%E a(49)-a(50) from _Benjamin Chaffin_, Dec 26 2009

%E a(51)-a(52) from _Benjamin Chaffin_, Jan 10 2010

%E a(53)-a(80) from _Benjamin Chaffin_, Jan 10 2012