1,2

Start with an initial string 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 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).

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.

F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and A. R. 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 A. R. Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence [pdf, ps].

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

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.

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

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

Cf. A091787, A090822, A093369, A094005, A116909.

Sequence in context: A073320 A020668 A020934 this_sequence A067271 A064394 A092022

Adjacent sequences: A094001 A094002 A094003 this_sequence A094005 A094006 A094007

nonn,more,nice,hard

N. J. A. Sloane (njas(AT)research.att.com), May 31 2004

a(27)-a(30) from Allan Wilks, Jul 29 2004

a(31)-a(36) from Benjamin Chaffin (chaffin(AT)gmail.com), Apr 09 2008