Fibonacci word

The first few elements of the infinite Fibonacci word are:

The critical exponent for the infinite Fibonacci word is repetitions.

The concatenation of two successive Fibonacci words is "almost commutative".

In analogy to its numerical counterpart, the Fibonacci word is defined by:

An example is the Fibonacci word.

The worst-case space usage of a suffix tree is seen with a fibonacci word, giving the full nodes.

Any prefix of the specific Fibonacci word belongs to L, but so do many other strings.

The infinite Fibonacci word is not periodic and not ultimately periodic.

The infinite Fibonacci word can contain repetitions of 3 successive identical subwords, but never 4.

A Fibonacci word is a specific sequence of binary digits (or symbols from any two-letter alphabet).

The infinite Fibonacci word is often cited as the worst case for algorithms detecting repetitions in a string.

The mathematical properties of the Fibonacci word and related topics are well researched and readily applied in such studies.

The infinite Fibonacci word is recurrent; that is, every subword occurs infinitely often.

As a consequence, the infinite Fibonacci word can be characterized by a cutting sequence of a line of slope or .

Alternatively, one can imagine directly generating the entire infinite Fibonacci word by the following process: start with a cursor pointing to the single digit 0.

However this sequence differs from the Fibonacci word only trivially, by swapping 0's for 1's and shifting the positions by one.

If is a subword of the infinite Fibonacci word, then so is its reversal, denoted .

The elements of a Fibonacci crystal structure are arranged in one or more spatial dimensions according to the sequence given by the Fibonacci word.

A famous example of (standard) Sturmian word is the Fibonacci word; its slope is , where is the golden ratio.

Suppressing the last two letters of a Fibonacci word, or prefixing the complement of the last two letters, creates a palindrome.

The Fibonacci word is formed by repeated concatenation in the same way that the Fibonacci numbers are formed by repeated addition.

The infinite Fibonacci word is a balanced sequence: Take two factors of the same length anywhere in the Fibonacci word.

Among aperiodic words, the largest possible palindromic density is achieved by the Fibonacci word, which has density 1/φ, where φ is the Golden ratio.

In the infinite Fibonacci word, the ratio (number of letters)/(number of zeroes) is φ, as is the ratio of zeroes to ones.

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.