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A191234 The number of strong binary words of length n. 0
2, 2, 4, 4, 8, 6, 12, 8, 12, 10, 16, 8, 12, 10, 16, 14, 20, 12, 18, 12, 20, 18, 26, 14, 20, 8, 12, 8, 12, 6, 10, 4, 6, 2, 4, 2, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Let s(w) denote the number of positions in a word that do not start a square. Then a word is said to be strong if for all nonempty prefixes u of w we have s(u)>=|u|/2.

The authors of the linked paper show that a(n)=0 for n>37, and thus all terms are known. For this reason the sequence is assigned the keyword "full" although it is actually not a finite sequence.

LINKS

Table of n, a(n) for n=1..37.

Tero Harju, Tomi Kärki and Dirk Nowotka, The Number of Positions Starting a Square in Binary Words, The Electronic Journal of Combinatorics, 18 (2011), #P6.

Eric Weisstein's World of Mathematics, Squarefree Word

EXAMPLE

Consider the binary word 0110 of length 4. The prefixes 0, 01, 011, and 0110 have 1, 2, 2, and 2 squarefree positions with ratios to length of 1, 1, 2/3, and 1/2, respectively. Since each ratio is greater than or equal to 1/2, 0110 is strong.

CROSSREFS

Sequence in context: A300123 A175359 A008330 * A225373 A138219 A279405

Adjacent sequences:  A191231 A191232 A191233 * A191235 A191236 A191237

KEYWORD

nonn,fini,full

AUTHOR

John W. Layman, May 27 2011

STATUS

approved

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Last modified March 22 12:46 EDT 2019. Contains 321421 sequences. (Running on oeis4.)