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A309077 Maximum sum of base lengths over all minimal factorizations of length-n binary strings. 2
1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 13, 13, 14, 14, 15, 16, 16, 17, 17, 18, 18, 19, 20, 20, 21, 22, 22, 23, 23, 24, 25, 25 (list; graph; refs; listen; history; text; internal format)
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

LINKS

Table of n, a(n) for n=1..40.

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

Cf. A309078.

Sequence in context: A073869 A060143 A005206 * A057365 A014245 A096386

Adjacent sequences:  A309074 A309075 A309076 * A309078 A309079 A309080

KEYWORD

nonn,more

AUTHOR

Jeffrey Shallit, Jul 11 2019

EXTENSIONS

a(21)-a(40) from Bert Dobbelaere, Jul 21 2019

STATUS

approved

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Last modified February 17 12:10 EST 2020. Contains 331996 sequences. (Running on oeis4.)