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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.
but these are not Zeckendorf representations because 34 and 21 are consecutive Fibonacci numbers, as are 5 and 3. For any given positive integer, its Zeckendorf representation can be found by using a greedy algorithm, choosing the largest possible Fibonacci number at each stage.
The sequence of the number of strings of 0s and 1s of length that contain at most consecutive 0s is also a Fibonacci sequence of order . These sequences, their limiting ratios, and the limit of these limiting ratios, were investigated by Mark Barr in 1913.
Dov Jarden proved that the Fibonomials appear as coefficients of an equation involving powers of consecutive Fibonacci numbers, namely Jarden proved that given any generalized Fibonacci sequence , that is, a sequence that satisfies = + for every , then
An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers F 25001 {\displaystyle F_{25001}} and F 25000 {\displaystyle F_{25000}} , each over 5000 {\displaystyle 5000} digits, yields over 10,000 {\displaystyle ...
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.
The sequence = / of ratios of consecutive Fibonacci numbers which, if it converges at all, converges to a limit satisfying = +, and no rational number has this property. If one considers this as a sequence of real numbers, however, it converges to the real number φ = ( 1 + 5 ) / 2 , {\displaystyle \varphi =(1+{\sqrt {5}})/2,} 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 ...