Abstract
We consider words over a binary alphabet. A word [Formula: see text] is overlap-free if it does not have factors (blocks of consecutive letters) of the form [Formula: see text] for nonempty [Formula: see text]. Let [Formula: see text] denote the number of positions that are middle positions of squares (of the form [Formula: see text]) in [Formula: see text]. We show that for overlap-free binary words, [Formula: see text], and that there are infinitely many overlap-free binary words for which [Formula: see text].
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