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The infinite Fibonacci word is the standard word generated by the directive sequence (1,1,1,....). The infinite Fibonacci word is recurrent; that is, every subword occurs infinitely often. If is a subword of the infinite Fibonacci word, then so is its reversal, denoted . If is a subword of the infinite Fibonacci word, then the least period of ...
The Fibonacci word fractal is a fractal curve defined on the plane from the Fibonacci word. Definition ... At the limit, the infinite Fibonacci tile tiles the plane.
A different trick may be used to prove = = + or in words, the sum of the squares of the first Fibonacci numbers up to is the product of the n-th and (n + 1)-th Fibonacci numbers. To see this, begin with a Fibonacci rectangle of size F n × F n + 1 {\displaystyle F_{n}\times F_{n+1}} and decompose it into squares of size F n , F n − 1 , . . .
The usual Fibonacci numbers are a Fibonacci sequence of order 2. The cases n = 3 {\displaystyle n=3} and n = 4 {\displaystyle n=4} have been thoroughly investigated. The number of compositions of nonnegative integers into parts that are at most n {\displaystyle n} is a Fibonacci sequence of order n {\displaystyle n} .
Hausdorff dimension (exact value) Hausdorff dimension (approx.) Name Illustration Remarks Calculated: 0.538: Feigenbaum attractor: The Feigenbaum attractor (see between arrows) is the set of points generated by successive iterations of the logistic map for the critical parameter value =, where the period doubling is infinite.
The Fibonacci word is an example of a Sturmian word. The start of the cutting sequence shown here illustrates the start of the word 0100101001.. In mathematics, a Sturmian word (Sturmian sequence or billiard sequence [1]), named after Jacques Charles François Sturm, is a certain kind of infinitely long sequence of characters.
A Sturmian word over a binary alphabet is one with complexity function n + 1. [14] A sequence is Sturmian if and only if it is balanced and aperiodic. [2] [15] An example is the Fibonacci word. [14] [16] More generally, a Sturmian word over an alphabet of size k is one with complexity n+k−1.
For example, the word "encyclopedia" is a sequence of symbols in the English alphabet, a finite set of twenty-six letters. Since a word can be described as a sequence, other basic mathematical descriptions can be applied. The alphabet is a set, so as one would expect, the empty set is a subset. In other words, there exists a unique word of ...