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Brasch et al. 2012 show how a generalized Fibonacci sequence also can be connected to the field of economics. [97] In particular, it is shown how a generalized Fibonacci sequence enters the control function of finite-horizon dynamic optimisation problems with one state and one control variable.
The term Fibonacci sequence is also applied more generally to any function from the integers to a field for which (+) = + (+).These functions are precisely those of the form () = () + (), so the Fibonacci sequences form a vector space with the functions () and () as a basis.
In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers. The polynomials generated in a similar way from the Lucas numbers are called Lucas polynomials .
For generalized Fibonacci sequences (satisfying the same recurrence relation, but with other initial values, e.g. the Lucas numbers) the number of occurrences of 0 per cycle is 0, 1, 2, or 4. The ratio of the Pisano period of n and the number of zeros modulo n in the cycle gives the rank of apparition or Fibonacci entry point of n.
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.
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.
For example, the problem of computing the Fibonacci sequence exhibits overlapping subproblems. The problem of computing the n th Fibonacci number F ( n ), can be broken down into the subproblems of computing F ( n − 1) and F ( n − 2), and then adding the two.
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 ...