Definition:Fibonacci Number

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Definition

The Fibonacci numbers are a sequence $\sequence {F_n}$ of integers which is formally defined recursively as:

$F_n = \begin {cases} 0 & : n = 0 \\ 1 & : n = 1 \\ F_{n - 1} + F_{n - 2} & : \text {otherwise} \end {cases}$

for all $n \in \Z_{\ge 0}$.

That is, the next integer in the sequence is found by adding together the two previous ones.


Negative

The definition of Fibonacci numbers for negative integers is an extension of the definition for positive integers:

$F_n = \begin{cases} 0 & : n = 0 \\ 1 & : n = 1 \\ F_{n + 2} - F_{n + 1} & : n < 0 \end{cases}$

for all $n \in \Z$.


Sequence

The sequence of Fibonacci numbers begins:

$0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \ldots$

This sequence is A000045 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Examples

Fibonacci Number $F_{173}$

The Fibonacci number $F_{173}$ is:

$638 \, 817 \, 435 \, 613 \, 190 \, 763 \, 972 \, 389 \, 505 \, 493$

It can be factorized as:

$6 \, 260 \, 874 \, 567 \, 713 \times 102 \, 033 \, 258 \, 884 \, 875 \, 061$


Fibonacci Number $F_{1000}$

The Fibonacci number $F_{1000}$ is a number with $209$ decimal digits beginning with $4$.


Also known as

According to some sources, this sequence is also known as Lamé's Sequence, after Gabriel Lamé.

However, this suggestion is difficult to corroborate.


The notation for the $n$th Fibonacci number is not universally standardised.

In much professional literature, $u_n$ is used.

$F_n$ tends to appear more in amateur publications, but $F_n$ is rapidly taking over as standard.

$F_n$ is the notation used in The Fibonacci Quarterly.


Also see

  • Results about Fibonacci numbers can be found here.


Source of Name

This entry was named for Leonardo Fibonacci.


Historical Note

Leonardo Fibonacci famously discussed this sequence in his Liber Abaci, in the context of breeding pairs of rabbits.

Hence the name Fibonacci numbers, which was given to this sequence by Édouard Lucas, who studied them in detail.


The sequence $\sequence {F_n}$ was known to Indian mathematicians as long ago as the $7$th century C.E.

It was also studied by Gopala before $1135$, and by Acharya Hemachandra in about $1150$.

Hence some sources refer to these numbers as the Gopala-Hemachandra numbers.

They are also discussed by Johannes Kepler in his work of $1611$ De Nive Sexangula (On the Six-Cornered Snowflake). It is suspected that Kepler was himself unfamiliar with Fibonacci's work.

Kepler himself had noticed the appearance of Fibonacci numbers in the growth of plants:

It is in the likeness of this self-developing series that the faculty of propagation is, in my opinion, formed; and so in a flower the authentic flag of this faculty is shown, the pentagon. I pass over all the other arguments that a delightful rumination could adduce in proof of this.


Linguistic Note

Fibonacci is an Italian surname whose pronunciation is something like Fib-bo-nat-chi, or Fib-bo-nar-chi, according to taste.

Avoid pronouncing it Fie-bo-nac-ky.


Sources