On the number of frames in binary words

A frame is a square uu, 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 bn/2c + 8 for all n and greater than 7n/30− for any positive 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 ThueMorse word, we prove that the minimum number of occurrences of frames in a word of length n is dn/2e − 2.