The exact number of squares in Fibonacci words

Abstract

All our words (sequences) are binary. A square is a subword of the form uu (concatenation). Two squares are distinct if they are of different shape, not just translates of each other. Otherwise they are repeated. Fibonacci words are defined by ƒ 0 = 0 , ƒ 1 = 1 , ƒ m = ƒ m − 1ƒ m − 2 for m ⩾ 2. Let F m = ¦ƒ m¦ . Then F 0 = 1, F 1 = 1, F m = F m − 1 + F m − 2 ( m ⩾ 2) are the Fibonacci numbers. Let D( n) and R( n) be the exact number of distinct and repeated squares, respectively, in ƒ n . We prove: D( n) = 2( F n − 2 − 1) ( n ⩾ 5), which implies, asymptotically, D( n) = 2(2 − ϑ) F n + o(1)(2(2 − ϑ) ≈ 0.7639), where ϑ is the golden section. We also prove: R(n) = 4 5 nF n − 2 5 (n + 6)F n − 1 − F n − 2 + n + 1 (n ⩾ 3) . This yields R(n) = 2 5 (3 − ϑ)nF n + O(F n) = (2(3 − ϑ) (5 log 2 ϑ))F n log 2F n + O(F n)(2(3 − ϑ) (5 log 2 ϑ) ≈ 0.7962) .