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Fibonacci numbers are used in a polyphase version of the merge sort algorithm in which an unsorted list is divided into two lists whose lengths correspond to sequential Fibonacci numbers—by dividing the list so that the two parts have lengths in the approximate proportion φ.
Plot of the first 10,000 Pisano periods. 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.
In reading Liber Abaci, it is helpful to understand Fibonacci's notation for rational numbers, a notation that is intermediate in form between the Egyptian fractions commonly used until that time and the vulgar fractions still in use today. [13] Fibonacci's notation differs from modern fraction notation in three key ways:
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.
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.
From a group and a field (or more generally a ring), the group ring [] is defined as the set of all finite formal -linear combinations of elements of − that is, an element of [] is of the form =, where = for all but finitely many so that the linear combination is finite.
F, also called the Fibonacci factorial, where n is a nonnegative integer, is defined as the product of the first n positive Fibonacci numbers, i.e. !:= =,, where F i is the i th Fibonacci number, and 0! F gives the empty product (defined as the multiplicative identity, i.e. 1).