OFFSET
1,3
COMMENTS
k is the "fractional curling number" of S(1),...,S(n). The infinite sequence S(1), S(2), ... is a fractional analog of Gijswijt's sequence A090822.
For the first 1000 terms, 1 <= S(n) <= 2. Is this always true?
The fractional curling number k of S = (S(1), S(2), ..., S(n)) is defined as follows. Write S = X Y Y ... Y Y' where X may be empty, Y is nonempty, there are say i copies of Y, and Y' is a prefix of Y. There may be many ways to do this. Choose the version in which the ratio k = (i|Y|+|Y'|)/|Y| is maximized; this k is the fractional curling number of S.
For example, if S = (S(1), ..., S(6)) = (1, 1, 2, 1, 3/2, 1), the best choice is to take X = 1,1,2, Y = 1,3/2, Y' = 1, giving k = (2+1)/2 = 3/2 = S(7).
LINKS
Allan Wilks, Table of n, a(n) for n = 1..10000 (terms 1..1000 from N. J. A. Sloane)
N. J. A. Sloane, Maple program for fractional curling number and S(1),S(2),...
Allan Wilks, Table of n, S(n) for n = 1..10000 [The first 1000 terms were computed by N. J. A. Sloane]
EXAMPLE
The sequence S(1), S(2), ... begins 1, 1, 2, 1, 3/2, 1, 3/2, 2, 6/5, 1, 5/4, 1, 3/2, 4/3, 1, 4/3, 3/2, 5/4, 8/7, 1, 6/5, 13/12, 1, 4/3, 5/4, 8/7, 9/7, 1, 6/5, 5/4, 6/5, 3/2, 16/15, 1, 7/6, 1, 3/2, 6/5, 8/7, 14/13, 1, 6/5, 5/4, 16/13, 1, 5/4, 4/3, 24/23, 1, 5/4, 3/2, 15/14, 1, 5/4, 3/2, 7/4, ...
MAPLE
See link.
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Conference dinner party, Workshop on Challenges in Combinatorics on Words, Fields Institute, Toronto, Apr 22 2013, entered by N. J. A. Sloane
STATUS
approved