A094004 - OEIS

A094004

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.

1, 4, 5, 8, 9, 14, 15, 66, 68, 70, 123, 124, 125, 132, 133, 134, 135, 136, 138, 139, 140, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 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

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OFFSET

1,2

COMMENTS

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.

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

LINKS

Table of n, a(n) for n=1..80.

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 [math.CO], 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.

Benjamin Chaffin and N. J. A. Sloane, The Curling Number Conjecture, preprint.

Index entries for sequences related to curling numbers

EXAMPLE

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.

CROSSREFS

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

Sequence in context: A073320 A020668 A020934 * A228012 A067271 A268128

Adjacent sequences: A094001 A094002 A094003 * A094005 A094006 A094007

KEYWORD