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Home > Fibonacci number

In words: you start with 0 and 1, and then produce the next Fibonacci number by adding the two previous Fibonacci numbers. The first Fibonacci numbers (sequence A000045 in OEIS) for n = 0, 1,... are

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144Integers CardinalOne hundred [and] forty four Ordinal144th Factorization Roman numeralCXLIV Binary10010000 Hexadecimal90 144 is the natural number following 143 and preceding 145. 144 is a dozen dozens, or one gross . It is the twelfth Fibonacci number, a, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946...

1 Origins

As documented by Knuth in The Art of Computer ProgrammingThe Art of Computer Programming is a comprehensive monograph written by Donald Knuth which covers all kinds of programming algorithms. The first three volumes are published, two others are planned. Originally, it was planned as a single volume of ten chap, this sequence was first described by the Indian mathematicians GopalaGopala was an Indian mathematician, who studied the so-called Fibonacci numbers, also called the Gopala-Hemachandra numbers, in 1135. See also Indian mathematicians External link Mathematicians Indian people. and HemachandraHemachandra Suri ( (correct Sanskrit spelling Hemacandra Sur ( 1089 1172) was one of the greatest scholars of his time. He wrote on many subjects: grammar, philosophy, tradition, and contemporary history. One of his best known works is the "Tri-shashthi-s - investigating the possible ways of exactly bin packing items of length 1 and 2. In the West, it was first studied by Leonardo of PisaLeonardo of Pisa or Leonardo Pisano (c. 1175 1250), also known as Fibonacci was an Italian mathematician and is best known for the invention of the Fibonacci numbers and his role in the introduction to Europe of the modern positional decimal system for wr, who was also known as Fibonacci (c. 1200), to describe the growth of a hypothetical rabbit population. The numbers describe the number of pairs in a (somewhat idealized) rabbit population after n months if it is assumed that

• the first month there is just one newly born pair,
• newly born pairs become productive from their second month on,
• we have no genetic problems whatsoever generated by inbreeding,
• each month every productive pair begets a new pair, and
• the rabbits never die

The formula above applies to the rabbit problem because if in month n we have a rabbits and in month n + 1 we have b rabbits then in month n + 2 we'll necessarily have a + b rabbits. That's because we know each rabbit basically gives birth to another each month (actually each pair gives birth to another pair, but it is the same thing) and that means that all a rabbits give birth to another number of a rabbits which will become fertile after two months, which is exactly at month n + 2. That's why we have the population at moment n + 1 (which is b) plus exactly the population at moment n (which is a).

The term Fibonacci sequence is also applied more generally to any functionIn mathematics, a function is a relation such that each element of a set (the domain is associated with a unique element of another (possibly the same) set (the codomain not to be confused with the range . The concept of a function is fundamental to virtu g where g(n + 2) = g(n) + g(n + 1). These functions are precisely those of the form g(n) = aF(n) + bF(n + 1) for some numbers a and b, so the Fibonacci sequences form a vector space with the functions F(n) and F(n + 1) as a basis.

In particular, the Fibonacci sequence L with L(1) = 1 and L(2) = 3 is referred to as the Lucas numbers. This sequence was described by Leonhard Euler in 1748, in the Introductio in Analysin Infinitorum. The significance in the Lucas numbers L(n) lies in the fact that raising the Golden ratio to the nth power yields:

Lucas numbers are related to Fibonacci numbers by the relation:

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Fibonacci numberIn mathematics, the Fibonacci numbers form a sequence defined recursively by: : In words: you start with 0 and 1, and then produce the next Fibonacci number by adding the two previous Fibonacci numbers. The first Fibonacci numbers (sequence A000045 in OEI Fibonacci codingThe Fibonacci code is a universal code which encodes positive integers into binary code words. All tokens end with "11" and have no "11" before the end. The code begins as follows: 1 11 2 011 3 0011 4 1011 5 00011 6 10011 7 01011 8 000011 9 100011 10 0100 Fibonacci heapTrees (structure) In computer science, a Fibonacci heap is a data structure similar to a binomial heap but with a better amortized running time. A Fibonacci heap can be used to improve the running time of Dijkstra's algorithm for computing shortest paths Fibonacci polynomialsIn mathematics, Fibonacci polynomials are a generalization of Fibonacci numbers. These polynomials are defined by: : The first few Fibonacci polynomials are: : : : : : : The Fibonacci numbers are recovered by evaluating the polynomials at x 1. See also Po Fibonacci number programIn many beginning computer science courses, an introduction to the concept of recursion often includes a program to calculate and print Fibonacci numbers (or computing the factorial of a number). In general, however, a recursive algorithm to compute Fibon Fibonacci pseudoprimeIn number theory, a pseudoprime is a number that passes some test that all primes pass, but is actually composite. A Fibonacci pseudoprime is a composite integer n that satisfies the following conditions: P > 0 and Q +1 or −1 V is congruent to P mod