cp's OEIS Frontend

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.

%I A217208 #33 Jul 08 2013 01:00:48
%S A217208 0,2,2,4,4,8,8,58,59,60,112,112,112,118,118,118,118,118,119,119,119,
%T A217208 120,120,120,120,120,120,120,120,120,120,120,120,120,120,120,120,120,
%U A217208 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
%N 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.
%C A217208 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:
%C A217208 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.
%C A217208 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 .
%C A217208 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 A217208 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.
%H A217208 F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">A Slow-Growing Sequence Defined by an Unusual Recurrence</a>, J. Integer Sequences, Vol. 10 (2007), #07.1.2. [<a href="http://neilsloane.com/doc/gijs.pdf">pdf</a>, <a href="http://neilsloane.com/doc/gijs.ps">ps</a>]
%H A217208 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. <a href="http://arxiv.org/abs/1212.6102">arXiv:1212.6102</a>
%H A217208 Benjamin Chaffin and N. J. A. Sloane, <a href="http://neilsloane.com/doc/CNC.pdf">The Curling Number Conjecture</a>, preprint.
%H A217208 <a href="/index/Cu#curling_numbers">Index entries for sequences related to curling numbers</a>
%e A217208 a(3) = 2, using the starting string 3,2,2, which extends to 3,2,2,2,3, of length 5.
%e A217208 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.
%e A217208 a(8) = 58: start = 23222323, end = 232223232223222322322232223232223222322322232223232223222322322332.
%e A217208 a(22) = 120: start = 2322322323222323223223: see A116909 for trajectory.
%Y A217208 a(n) = length of n-th row of A217209.
%Y A217208 a(n) = A094004(n) - n.
%Y A217208 Cf. A091787, A090822, A093369, A094005, A116909, A160766, A216730, A216813.
%K A217208 nonn,nice,hard
%O A217208 1,2
%A A217208 _N. J. A. Sloane_, Sep 29 2012; revised Oct 02 2012