Fibonacci word

A Fibonacci word is a specific sequence of binary digits (or symbols from any two-letter alphabet). The Fibonacci word is formed by repeated concatenation in the same way that the Fibonacci numbers are formed by repeated addition. A Fibonacci word is a specific sequence of binary digits (or symbols from any two-letter alphabet). The Fibonacci word is formed by repeated concatenation in the same way that the Fibonacci numbers are formed by repeated addition. It is a paradigmatic example of a Sturmian word and specifically, a morphic word. The name “Fibonacci word” has also been used to refer to the members of a formal language L consisting of strings of zeros and ones with no two repeated ones. Any prefix of the specific Fibonacci word belongs to L, but so do many other strings. L has a Fibonacci number of members of each possible length. Let S 0 {displaystyle S_{0}} be '0' and S 1 {displaystyle S_{1}} be '01'. Now S n = S n − 1 S n − 2 {displaystyle S_{n}=S_{n-1}S_{n-2}} (the concatenation of the previous sequence and the one before that). The infinite Fibonacci word is the limit S ∞ {displaystyle S_{infty }} , that is, the (unique) infinite sequence that contains each S n {displaystyle S_{n}} , for finite n {displaystyle n} , as a prefix.