Factorizations of the Fibonacci Infinite Word

The aim of this note is to survey the factorizations of the Fibonacci infinite word that make use of the Fibonacci words and other related words, and to show that all these factorizations can be easily derived in sequence starting from elementary properties of the Fibonacci numbers. 1 Preliminaries The well-known sequence of Fibonacci numbers (sequence A000045 in the On-Line Encyclopedia of Integer Sequences) is defined by F1 = 1, F2 = 1 and for every n > 2, Fn = Fn+1 + Fn+2. The first few values of the sequence Fn are reported in Table 1 for reference. n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Fn 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 Table 1: The first few values of the sequence of Fibonacci numbers. A basic property of Fibonacci numbers (that can be easily proved by induction) is that 1 plus the sum of the first n Fibonacci numbers is equal to the (n+2)-th Fibonacci number: