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  2. 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. ... non-piecewise formula, ...

  3. 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 ... An alternate recursive formula for the limit of ... a non-profit ...

  4. Recurrence relation - Wikipedia

    en.wikipedia.org/wiki/Recurrence_relation

    In mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only previous terms of the sequence appear in the equation, for a parameter that is independent of ; this number is called the order of the relation.

  5. 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.

  6. Constant-recursive sequence - Wikipedia

    en.wikipedia.org/wiki/Constant-recursive_sequence

    The Fibonacci sequence is constant-recursive: each element of the sequence is the sum of the previous two. Hasse diagram of some subclasses of constant-recursive sequences, ordered by inclusion In mathematics , an infinite sequence of numbers s 0 , s 1 , s 2 , s 3 , … {\displaystyle s_{0},s_{1},s_{2},s_{3},\ldots } is called constant ...

  7. Generating function - Wikipedia

    en.wikipedia.org/wiki/Generating_function

    The ordinary generating function of a sequence can be expressed as a rational function (the ratio of two finite-degree polynomials) if and only if the sequence is a linear recursive sequence with constant coefficients; this generalizes the examples above. Conversely, every sequence generated by a fraction of polynomials satisfies a linear ...

  8. Zeckendorf's theorem - Wikipedia

    en.wikipedia.org/wiki/Zeckendorf's_theorem

    where F n is the n th Fibonacci number. Such a sum is called the Zeckendorf representation of N. The Fibonacci coding of N can be derived from its Zeckendorf representation. For example, the Zeckendorf representation of 64 is 64 = 55 + 8 + 1. There are other ways of representing 64 as the sum of Fibonacci numbers 64 = 55 + 5 + 3 + 1 64 = 34 ...

  9. Fibonomial coefficient - Wikipedia

    en.wikipedia.org/wiki/Fibonomial_coefficient

    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