On the number of frames in binary words

Abstract A frame is a square u u , where u is an unbordered word. Let F ( n ) denote the maximum number of distinct frames in a binary word of length n . We count this number for small values of n and show that F ( n ) is at most ⌊ n / 2 ⌋ + 8 for all n and greater than 7 n / 30 − ϵ for any positive ϵ and infinitely many  n . We also show that Fibonacci words, which are known to contain plenty of distinct squares, have only a few frames. Moreover, by modifying the Thue–Morse word, we prove that the minimum number of occurrences of frames in a word of length  n is ⌈ n / 2 ⌉ − 2 .

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