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Fibonacci numbers are also strongly related to the golden ratio: Binet's formula expresses the n-th Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases. Fibonacci numbers are also closely related to Lucas numbers, which obey the same ...
The ratio between two consecutive elements converges to the golden ratio, except in the case of the sequence which is constantly zero and the sequences where the ratio of the two first terms is (). The sequence can be written in the form
Exceptionally, the golden ratio is equal to the limit of the ratios of successive terms in the Fibonacci sequence and sequence of Lucas numbers: [42] + = + =. In other words, if a Fibonacci and Lucas number is divided by its immediate predecessor in the sequence, the quotient approximates φ {\displaystyle \varphi } .
The reciprocal Fibonacci constant ψ is the sum of the reciprocals of the Fibonacci numbers: = = = + + + + + + + +. Because the ratio of successive terms tends to the reciprocal of the golden ratio, which is less than 1, the ratio test shows that the sum converges.
The sequence also has a variety of relationships with the Fibonacci numbers, like the fact that adding any two Fibonacci numbers two terms apart in the Fibonacci sequence results in the Lucas number in between. [3] The first few Lucas numbers are 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, ... .
Fibonacci search has an average- and worst-case complexity of O(log n) (see Big O notation). The Fibonacci sequence has the property that a number is the sum of its two predecessors. Therefore the sequence can be computed by repeated addition. The ratio of two consecutive numbers approaches the Golden ratio, 1.618... Binary search works by ...
The sequence 2, 1, 3, 4, 7, 11, ... of Lucas numbers satisfies the same recurrence as the Fibonacci sequence but with initial conditions = and =. More generally, every Lucas sequence is constant-recursive of order 2. [2]
[3] In the infinite Fibonacci word, the ratio (number of letters)/(number of zeroes) is φ, as is the ratio of zeroes to ones. [4] The infinite Fibonacci word is a balanced sequence: Take two factors of the same length anywhere in the Fibonacci word. The difference between their Hamming weights (the number of occurrences of "1") never exceeds 1 ...