Fibonacci Word

# Fibonacci word

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Description:
thumb|350px|Characterization by a cutting sequence with a line of slope [itex]varphi[/itex] or [itex]varphi[/itex]-1 with [itex]varphi[/itex], the 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.

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.

## Definition

Let [itex]S_0[/itex] be "0" and [itex]S_1[/itex] be "01". Now [itex]S_n = S_S_[/itex] (the concatenation of the previous sequence and the one before that).

The infinite Fibonacci word is the limit [itex]S_[/itex].

## The Fibonacci words

We have:

[itex]S_0[/itex] &nbsp;&nbsp; 0

[itex]S_1[/itex] &nbsp;&nbsp; 01

[itex]S_2[/itex] &nbsp;&nbsp; 010

[itex]S_3[/itex] &nbsp;&nbsp; 01001

[itex]S_4[/itex] &nbsp;&nbsp; 01001010

[itex]S_5[/itex] &nbsp;&nbsp; 0100101001001

...

The first few elements of the infinite Fibonacci word are:

0,......

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