The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

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