A093370 Start with any initial string of n numbers s(1), ..., s(n), with s(1) = 2, other s(i)'s = 2 or 3 (so there are 2^(n-1) starting strings). The rule for extending the string is this as follows: To get s(n+1), write the string s(1)s(2)...s(n) 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. Then a(n) = number of starting strings for which k > 1.

0, 1, 2, 5, 10, 22, 44, 91, 182, 369, 738, 1486, 2972, 5962, 11924, 23884, 47768, 95607, 191214, 382568, 765136, 1530552, 3061104, 6122765, 12245530, 24492171, 48984342, 97970902, 195941804, 391888040

Offset

1,3

Links

Table of n, a(n) for n=1..30.

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.

Index entries for sequences related to Gijswijt's sequence

Index entries for sequences related to curling numbers

Formula

Equals A121880(n)/2, or 2^(n-1) - A122536(n)/2.

a(n)/2^(n-1) seems to converge to a number around 0.73.

Example

For n=2 there are 2 starting strings, 22 and 23 and only the first has k > 1.
For n=4 there are 8 starting strings, but only 5 have k > 1, namely 2222, 2233, 2322, 2323, 2333.

Crossrefs

Cf. A090822, A093369, A093371, A121880, A122536.

Sequence in context: A073777 A215422 A026633 * A372957 A094537 A135098

Adjacent sequences: A093367 A093368 A093369 * A093371 A093372 A093373

Keyword

nonn

Author

N. J. A. Sloane, Apr 28 2004

Extensions

More terms from Guy P. Srinivasan, via A122536, Sep 18 2006

Status

approved