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The number of binary strings of length n without an odd number of consecutive 1 s is the Fibonacci number F n+1. For example, out of the 16 binary strings of length 4, there are F 5 = 5 without an odd number of consecutive 1 s—they are 0000, 0011, 0110, 1100, 1111.
The sequence of Fibonacci strings starts: b, a, ab, aba, abaab, abaababa, abaababaabaab, … (sequence A106750 in the OEIS) The length of each Fibonacci string is a Fibonacci number, and similarly there exists a corresponding Fibonacci string for each Fibonacci number. Fibonacci strings appear as inputs for the worst case in some computer ...
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.
where F n is the n th Fibonacci number. 1974 [OEIS 74] ? ? ... [λ n,2] is the expected longest common subsequence of two random length-n binary strings 1975 ? ? ?
The Young–Fibonacci graph is an infinite graph, with a vertex for each string of the digits "1" and "2" (including the empty string). The neighbors of a string s are the strings formed from s by one of the following operations: Insert a "1" into s, prior to the leftmost "1" (or anywhere in s if it does not already contain a "1").
In the Fibonacci sequence, each number is the sum of the previous two numbers. Fibonacci omitted the "0" and first "1" included today and began the sequence with 1, 2, 3, ... . He carried the calculation up to the thirteenth place, the value 233, though another manuscript carries it to the next place, the value 377.
Taking these in their natural order gives the sequence (2, 3, 5, 7, 11, 13, 17, ...). The prime numbers are widely used in mathematics, particularly in number theory where many results related to them exist. The Fibonacci numbers comprise the integer sequence in which each element is the sum of the previous two elements. The first two elements ...
For example, in the Fibonacci arithmetic system, based on the Fibonacci sequence, the number 17 can be encoded in six different ways: 110111 (F 6 + F 5 + F 3 + F 2 + F 1 = 8 + 5 + 2 + 1 + 1 = 17, maximal form)