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A330881 Length of longest LB factorization over all binary strings of length n. 3
0, 1, 2, 3, 4, 5, 5, 5, 6, 7, 8, 9, 10, 11, 11, 11, 12, 13, 14, 15, 16, 17, 17, 17, 18, 19, 20, 21, 22, 23, 23, 23, 24, 25, 26, 27, 28, 29, 29, 29, 30, 31, 32, 33, 34, 35, 35, 35, 36, 37, 38, 39, 40, 41, 41, 41, 42, 43, 44, 45, 46, 47, 47, 47, 48, 49, 50, 51, 52 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A border of a string w is a nonempty proper prefix of w that is also a suffix.  The LB ("longest border") factorization of a string w is as follows:  if w has no border, then the factorization is just (w).  Otherwise, write w = (x)(w')(x) where x is the longest border of length <= |w|/2, and continue with w'.   The length of the factorization is the number of factors.  For example, 0101101010 = (010)(1)(10)(1)(010), and so has length 5.

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-2,2,-1).

FORMULA

For n >= 0, we have a(8n+i) = 6n+i for 0 <= i <= 5, and a(8n+6) = a(8n+7) = 6n+5.

From Colin Barker, May 01 2020: (Start)

G.f.: x*(1 + x^2 + x^4 - x^5 + x^6) / ((1 - x)^2*(1 + x^2)*(1 + x^4)).

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - 2*a(n-6) + 2*a(n-7) - a(n-8) for n>7.

(End)

EXAMPLE

For n = 7 an example achieving a(7) = 5 is 0101100 = (0)(10)(1)(10)(0).

PROG

(PARI) concat(0, Vec((1 + x^2 + x^4 - x^5 + x^6)/((-1 + x)^2*(1 + x^2 + x^4 + x^6)) + O(x^70))) \\ Jinyuan Wang, May 01 2020

CROSSREFS

Cf. A330882, A330884.

Sequence in context: A200262 A270432 A007599 * A279317 A154940 A133344

Adjacent sequences:  A330878 A330879 A330880 * A330882 A330883 A330884

KEYWORD

nonn,easy

AUTHOR

Jeffrey Shallit, Apr 30 2020

EXTENSIONS

More terms from Jinyuan Wang, May 01 2020

STATUS

approved

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Last modified July 30 17:21 EDT 2021. Contains 346359 sequences. (Running on oeis4.)