Fibonacci word

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Description:
thumb|350px|Characterization by a cutting sequence with a line of slope <math>varphi</math> or <math>varphi</math>-1 with <math>varphi</math>, 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 <math>S_0</math> be "0" and <math>S_1</math> be "01". Now <math>S_n = S_S_</math> (the concatenation of the previous sequence and the one before that).

The infinite Fibonacci word is the limit <math>S_</math>.

The Fibonacci words

We have:

<math>S_0</math> &nbsp;&nbsp; 0

<math>S_1</math> &nbsp;&nbsp; 01

<math>S_2</math> &nbsp;&nbsp; 010

<math>S_3</math> &nbsp;&nbsp; 01001

<math>S_4</math> &nbsp;&nbsp; 01001010

<math>S_5</math> &nbsp;&nbsp; 0100101001001

...

The first few elements of the infinite Fibonacci word are:

0,......
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