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A105098 Minimum number of squares (repeated adjacent nonempty subwords) in a binary string of length n. 1
0, 0, 0, 1, 1, 2, 2, 2, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 20, 20 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

This sequence counts "occurrences" of (possibly identical) squares, overlaps allowed. See Kucherov et al. link. - Michael S. Branicky, Dec 20 2020

LINKS

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

G. Kucherov, P. Ochem and M. Rao, How many square occurrences must a binary sequence contain?, Electronic Journal of Combinatorics V. 10 (2003), paper #R12.

EXAMPLE

a(4) = 1 because every string of length > 3 contains at least one square and 0100 contains only one square.

PROG

(Python)

from itertools import product

def count_overlaps(subs, s):

c = i = 0

while i != -1:

i = s.find(subs, i)

if i != -1: c += 1; i += 1

return c

def a(n):

if n == 1: return 0

squares = ["".join(u) + "".join(u)

for r in range(1, n//2 + 1) for u in product("01", repeat = r)]

words = ("0"+"".join(w) for w in product("10", repeat=n-1))

themin = n*n

for w in words:

numw = 0

for s in squares:

numw += count_overlaps(s, w)

if numw >= themin: break

else: themin = min(themin, numw)

return themin

print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Dec 20 2020

CROSSREFS

Sequence in context: A300076 A029113 A244988 * A303003 A063124 A070547

Adjacent sequences: A105095 A105096 A105097 * A105099 A105100 A105101

KEYWORD

nonn,hard,more

AUTHOR

Jeffrey Shallit, Apr 07 2005

STATUS

approved

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Last modified November 29 10:31 EST 2022. Contains 358424 sequences. (Running on oeis4.)