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Every sequence of positive integers satisfying the Fibonacci recurrence occurs, shifted by at most finitely many positions, in the Wythoff array. In particular, the Fibonacci sequence itself is the first row, and the sequence of Lucas numbers appears in shifted form in the second row (Morrison 1980).
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.
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:
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.
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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.
To encode an integer N: . Find the largest Fibonacci number equal to or less than N; subtract this number from N, keeping track of the remainder.; If the number subtracted was the i th Fibonacci number F(i), put a 1 in place i − 2 in the code word (counting the left most digit as place 0).