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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 ...
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 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
The palindromic density of the infinite Fibonacci word is thus 1/φ, where φ is the golden ratio: this is the largest possible value for aperiodic words. [ 3 ] In the infinite Fibonacci word, the ratio (number of letters)/(number of zeroes) is φ, as is the ratio of zeroes to ones.
A Fibonacci spiral approximates the golden spiral using quarter-circle arcs inscribed in squares derived from the Fibonacci sequence. A golden spiral with initial radius 1 is the locus of points of polar coordinates ( r , θ ) {\displaystyle (r,\theta )} satisfying r = φ 2 θ / π , {\displaystyle r=\varphi ^{2\theta /\pi },} where φ ...
This characterization is exact: every sequence of complex numbers that can be written in the above form is constant-recursive. [20] For example, the Fibonacci number is written in this form using Binet's formula: [21] =,
Since the inverse of a metallic mean is less than 1, this formula implies that the quotient of two consecutive elements of such a sequence tends to the metallic mean, when k tends to the infinity. For example, if n = 1 , {\displaystyle n=1,} S n {\displaystyle S_{n}} is the golden ratio .
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 ...