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In mathematics, the Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers , commonly denoted F n .
The Fibonacci sequence is frequently referenced in the 2001 book The Perfect Spiral by Jason S. Hornsby. A youthful Fibonacci is one of the main characters in the novel Crusade in Jeans (1973). He was left out of the 2006 movie version, however. The Fibonacci sequence and golden ratio are briefly described in John Fowles's 1985 novel A Maggot.
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. [12] Fibonacci's notation differs from modern fraction notation in three key ways:
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.
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.
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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.
A digit sequence with rank r may be formed either by adding the digit 2 to a sequence with rank r − 2, or by adding the digit 1 to a sequence with rank r − 1.If f is the function that maps r to the number of different digit sequences of that rank, therefore, f satisfies the recurrence relation f (r) = f (r − 2) + f (r − 1) defining the Fibonacci numbers, but with slightly different ...