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  2. Generalizations of Fibonacci numbers - Wikipedia

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

    The semi-Fibonacci sequence (sequence A030067 in the OEIS) is defined via the same recursion for odd-indexed terms (+) = + and () =, but for even indices () = (), . The bisection A030068 of odd-indexed terms s ( n ) = a ( 2 n − 1 ) {\displaystyle s(n)=a(2n-1)} therefore verifies s ( n + 1 ) = s ( n ) + a ( n ) {\displaystyle s(n+1)=s(n)+a(n ...

  3. Fibonacci sequence - Wikipedia

    en.wikipedia.org/wiki/Fibonacci_sequence

    In mathematics, the Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers , commonly denoted F n .

  4. Fibonacci search technique - Wikipedia

    en.wikipedia.org/wiki/Fibonacci_search_technique

    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 ...

  5. Overlapping subproblems - Wikipedia

    en.wikipedia.org/wiki/Overlapping_subproblems

    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.

  6. Dynamic programming - Wikipedia

    en.wikipedia.org/wiki/Dynamic_programming

    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 ...

  7. Random Fibonacci sequence - Wikipedia

    en.wikipedia.org/wiki/Random_Fibonacci_sequence

    A random Fibonacci sequence is an integer random sequence given by the numbers for natural numbers, where = = and the subsequent terms are chosen randomly according to the random recurrence relation = {+,;,. An instance of the random Fibonacci sequence starts with 1,1 and the value of the each subsequent term is determined by a fair coin toss: given two consecutive elements of the sequence ...

  8. Formulas for generating Pythagorean triples - Wikipedia

    en.wikipedia.org/wiki/Formulas_for_generating...

    In this section we shall use the Fibonacci Box in place of the primitive triple it represents. An infinite ternary tree containing all primitive Pythagorean triples/Fibonacci Boxes can be constructed by the following procedure. [10] Consider a Fibonacci Box containing two, odd, coprime integers x and y in the right-hand column.

  9. Lucas number - Wikipedia

    en.wikipedia.org/wiki/Lucas_number

    The Lucas sequence has the same recursive relationship as the Fibonacci sequence, where each term is the sum of the two previous terms, but with different starting values. [1] This produces a sequence where the ratios of successive terms approach the golden ratio, and in fact the terms themselves are roundings of integer powers of the golden ...