A093369 - OEIS

%I #20 Jul 25 2017 08:15:53

%S 1,6,14,42,98,242,552,1394,2935,6471,14006,30060,64223,136914,290224,

%T 613509,1292567,2717311,5696864,11920124,24889066,51880008,107954163,

%U 224305440,465388743,964349526,1995808823,4125871527,8520180124,17577302639,36228352911

%N a(n) = sum of lengths of strings that can be generated by any starting string of n 2's and 3's that starts with a 2, using the rule described in the Comments lines.

%C 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:

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

%C a(n) = sum of final length of string, summed over all 2^(n-1) starting strings.

%H Lars Blomberg, <a href="/A093369/b093369.txt">Table of n, a(n) for n = 1..37</a>

%H 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.

%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, 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 <a href="/index/Ge#Gijswijt">Index entries for sequences related to Gijswijt's sequence</a>

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

%e a(3) = 14: the starting string, final string and length are as follows:

%e 222 2223 4

%e 223 223 3

%e 232 232 3

%e 233 2332 4, for a total of 4+3+3+4 = 14.

%Y Cf. A090822, A093370, A093371, A094004, A094005.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Apr 28 2004

%E a(21)-a(31) from _Lars Blomberg_, Jul 25 2017