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Brasch et al. 2012 show how a generalized Fibonacci sequence also can be connected to the field of economics. [96] 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.
The codenominator is a function that extends the Fibonacci sequence to the index set of positive rational numbers, +. Many known Fibonacci identities carry over to the codenominator. One can express Dyer's outer automorphism of the extended modular group PGL(2, Z) in terms of the codenominator.
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]
In the Fibonacci sequence, each number is the sum of the previous two numbers. Fibonacci omitted the "0" and first "1" included today and began the sequence with 1, 2, 3, ... . He carried the calculation up to the thirteenth place, the value 233, though another manuscript carries it to the next place, the value 377.
The Fibonacci and Lucas numbers are recovered by evaluating the polynomials at x = 1; Pell numbers are recovered by evaluating F n at x = 2. The ordinary generating functions for the sequences are: [ 4 ]
is a function, where X is a set to which the elements of a sequence must belong. For any , this defines a unique sequence with as its first element, called the initial value. [1] It is easy to modify the definition for getting sequences starting from the term of index 1 or higher.
There is a general method for expressing the general term of such a sequence as a function of n; see Linear recurrence. In the case of the Fibonacci sequence, one has =, = =, and the resulting function of n is given by Binet's formula.