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  2. Fibonacci sequence - Wikipedia

    en.wikipedia.org/wiki/Fibonacci_sequence

    5.3 Induction proofs. ... the Fibonacci sequence is a sequence in which each element is the ... A one-dimensional optimization method, called the Fibonacci search ...

  3. Zeckendorf's theorem - Wikipedia

    en.wikipedia.org/wiki/Zeckendorf's_theorem

    The first part of Zeckendorf's theorem (existence) can be proven by induction. 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 .

  4. Mathematical induction - Wikipedia

    en.wikipedia.org/wiki/Mathematical_induction

    Mathematical induction is a method for proving that a statement () is true for every natural number, that is, that the infinitely many cases (), (), (), (), … all hold. This is done by first proving a simple case, then also showing that if we assume the claim is true for a given case, then the next case is also true.

  5. Cassini and Catalan identities - Wikipedia

    en.wikipedia.org/wiki/Cassini_and_Catalan_identities

    A quick proof of Cassini's identity may be given (Knuth 1997, p. 81) by recognising the left side of the equation as a determinant of a 2×2 matrix of Fibonacci numbers. The result is almost immediate when the matrix is seen to be the n th power of a matrix with determinant −1:

  6. Greedy algorithm for Egyptian fractions - Wikipedia

    en.wikipedia.org/wiki/Greedy_algorithm_for...

    The expansion produced by this method for a number is called the greedy Egyptian expansion, Sylvester expansion, or Fibonacci–Sylvester expansion of . However, the term Fibonacci expansion usually refers, not to this method, but to representation of integers as sums of Fibonacci numbers .

  7. Generalizations of Fibonacci numbers - Wikipedia

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

    A Fibonacci sequence of order n is an integer sequence in which each sequence element is the sum of the previous elements (with the exception of the first elements in the sequence). The usual Fibonacci numbers are a Fibonacci sequence of order 2.

  8. Liber Abaci - Wikipedia

    en.wikipedia.org/wiki/Liber_Abaci

    Although the resulting Fibonacci sequence dates back long before Leonardo, [9] its inclusion in his book is why the sequence is named after him today. The fourth section derives approximations, both numerical and geometrical, of irrational numbers such as square roots. [10] The book also includes proofs in Euclidean geometry. [11]

  9. Binomial coefficient - Wikipedia

    en.wikipedia.org/wiki/Binomial_coefficient

    If one denotes by F(i) the sequence of Fibonacci numbers, indexed so that F(0) = F(1) = 1, then the identity = ⌊ ⌋ = has the following combinatorial proof. [12] One may show by induction that F ( n ) counts the number of ways that a n × 1 strip of squares may be covered by 2 × 1 and 1 × 1 tiles.