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  2. Fibonacci sequence - Wikipedia

    en.wikipedia.org/wiki/Fibonacci_sequence

    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 ...

  3. Generalizations of Fibonacci numbers - Wikipedia

    en.wikipedia.org/wiki/Generalizations_of...

    The usual Fibonacci numbers are a Fibonacci sequence of order 2. The cases n = 3 {\displaystyle n=3} and n = 4 {\displaystyle n=4} have been thoroughly investigated. The number of compositions of nonnegative integers into parts that are at most n {\displaystyle n} is a Fibonacci sequence of order n {\displaystyle n} .

  4. Fibonacci search technique - Wikipedia

    en.wikipedia.org/wiki/Fibonacci_search_technique

    Let k be defined as an element in F, the array of Fibonacci numbers. n = F m is the array size. If n is not a Fibonacci number, let F m be the smallest number in F that is greater than n. The array of Fibonacci numbers is defined where F k+2 = F k+1 + F k, when k ≥ 0, F 1 = 1, and F 0 = 1. To test whether an item is in the list of ordered ...

  5. Recurrence relation - Wikipedia

    en.wikipedia.org/wiki/Recurrence_relation

    A famous example is the recurrence for the Fibonacci numbers, = + where the order is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients , because the coefficients of the linear function (1 and 1) are constants that do not depend on n . {\displaystyle n.}

  6. Fibonacci cube - Wikipedia

    en.wikipedia.org/wiki/Fibonacci_cube

    The Fibonacci cube of order n is the simplex graph of the complement graph of an n-vertex path graph. [2] That is, each vertex in the Fibonacci cube represents a clique in the path complement graph, or equivalently an independent set in the path itself; two Fibonacci cube vertices are adjacent if the cliques or independent sets that they represent differ by the addition or removal of a single ...

  7. Golden-section search - Wikipedia

    en.wikipedia.org/wiki/Golden-section_search

    In order to approximate the probe positions of golden section search while probing only integer sequence indices, the variant of the algorithm for this case typically maintains a bracketing of the solution in which the length of the bracketed interval is a Fibonacci number.

  8. Zeckendorf's theorem - Wikipedia

    en.wikipedia.org/wiki/Zeckendorf's_theorem

    For n = 1, 2, 3 it is clearly true (as these are Fibonacci numbers), for n = 4 we have 4 = 3 + 1. If n is a Fibonacci number then there is nothing to prove. Otherwise there exists j such that F j < n < F j + 1 . Now suppose each positive integer a < n has a Zeckendorf representation (induction hypothesis) and consider b = n − F j .

  9. Pisano period - Wikipedia

    en.wikipedia.org/wiki/Pisano_period

    In number theory, the nth Pisano period, written as π (n), is the period with which the sequence of Fibonacci numbers taken modulo n repeats. Pisano periods are named after Leonardo Pisano, better known as Fibonacci. The existence of periodic functions in Fibonacci numbers was noted by Joseph Louis Lagrange in 1774. [1] [2]