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For example, consider the recursive formulation for generating the Fibonacci sequence: F i = F i−1 + F i−2, with base case F 1 = F 2 = 1. Then F 43 = F 42 + F 41, and F 42 = F 41 + F 40. Now F 41 is being solved in the recursive sub-trees of both F 43 as well as F 42. Even though the total number of sub-problems is actually small (only 43 ...
The n-Fibonacci constant is the ratio toward which adjacent -Fibonacci numbers tend; it is also called the n th metallic mean, and it is the only positive root of =. For example, the case of n = 1 {\displaystyle n=1} is 1 + 5 2 {\displaystyle {\frac {1+{\sqrt {5}}}{2}}} , or the golden ratio , and the case of n = 2 {\displaystyle n=2} is 1 + 2 ...
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 recursive program above is tail-recursive; it is equivalent to an iterative algorithm, and the computation shown above shows the steps of evaluation that would be performed by a language that eliminates tail calls. Below is a version of the same algorithm using explicit iteration, suitable for a language that does not eliminate tail calls.
A Fibonacci prime is a Fibonacci number that is prime. The first few are: [46] 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ... Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many. [47] F kn is divisible by F n, so, apart from F 4 = 3, any Fibonacci prime must have a prime index.
A recursive step — a set of rules that reduces all successive cases toward the base case. For example, the following is a recursive definition of a person's ancestor. One's ancestor is either: One's parent (base case), or; One's parent's ancestor (recursive step). The Fibonacci sequence is another classic example of recursion: Fib(0) = 0 as ...
Therefore, the computation of F(n − 2) is reused, and the Fibonacci sequence thus exhibits overlapping subproblems. A naive recursive approach to such a problem generally fails due to an exponential complexity. If the problem also shares an optimal substructure property, dynamic programming is a good way to work it out.
Recursion allows direct implementation of functionality defined by mathematical induction and recursive divide and conquer algorithms. Here is an example of a recursive function in C/C++ to find Fibonacci numbers: