OFFSET
1,2
COMMENTS
A factorization of a binary string x is an expression of the form prod_i w_i^{e_i}, where each w_i is a word and e_i is an integer exponent specifying how many times the word is repeated. For example 0101000 = (01)^2 0^3. A minimal factorization is one that minimizes the weight of the factorization, which is defined to be sum of the lengths of the w_i. a(n) then measures the maximum weight over all length-n binary strings.
Since there are arbitrarily long binary words having no repetitions larger than squares (Thue 1906), we see that a(n) >= n/2. By considering a(14) = 9, and splitting a word into blocks of size 14 and one left over, we see that a(n) <= 0.644 n for sufficiently large n.
Upper bound for sufficiently large n reduced to a(n) < 0.621 n considering a(29) = 18. - Bert Dobbelaere, Jul 21 2019
FORMULA
a(j+k) <= a(j) + a(k). - Bert Dobbelaere, Jul 21 2019
EXAMPLE
For n = 8, we have a(8) = 6, and a word that achieves the maximum is 01001101, where the corresponding weight-6 factorization is (01) 0^2 1^2 (01).
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Jeffrey Shallit, Jul 11 2019
EXTENSIONS
a(21)-a(40) from Bert Dobbelaere, Jul 21 2019
STATUS
approved