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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 idea of factoring of large numbers can be applied to words, where a factor of a word is a block of consecutive symbols. [1] Thus, "cyclop" is a factor of "encyclopedia". In addition to examining sequences in themselves, another area to consider of combinatorics on words is how they can be represented visually.
An alternate recursive formula for the limit of ratio of two consecutive -nacci numbers can be expressed as r = ∑ k = 0 n − 1 r − k {\displaystyle r=\sum _{k=0}^{n-1}r^{-k}} . The special case n = 2 {\displaystyle n=2} is the traditional Fibonacci series yielding the golden section φ = 1 + 1 φ {\displaystyle \varphi =1+{\frac {1 ...
This section features terms used across different areas in mathematics, or terms that do not typically appear in more specialized glossaries. For the terms used only in some specific areas of mathematics, see glossaries in Category:Glossaries of mathematics.
Equivalently, F n+2 is the number of subsets S of {1, ..., n} without consecutive integers, that is, those S for which {i, i + 1} ⊈ S for every i. A bijection with the sums to n+1 is to replace 1 with 0 and 2 with 10, and drop the last zero. The number of binary strings of length n without an odd number of consecutive 1 s is the Fibonacci ...
When the sizes of groups of consecutive terms grow without bounds, it is necessary to look at the behavior of .. Mirroring permutations and circular shift permutations, as well as their inverses, add at most 1 interval to the main interval [,], hence and its inverse are Agnew permutations (with =), i.e., mirroring and circular shifting can be applied within the groups with the convergence type ...
Gilbreath's conjecture is a conjecture in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the results unsigned, and then repeating this process on consecutive terms in the resulting sequence, and so forth.
A random Fibonacci sequence is an integer random sequence given by the numbers for natural numbers, where = = and the subsequent terms are chosen randomly according to the random recurrence relation = {+,;,. An instance of the random Fibonacci sequence starts with 1,1 and the value of the each subsequent term is determined by a fair coin toss: given two consecutive elements of the sequence ...