Fibonacci number

Fibonacci spiral

Fibonacci numbers form a sequence of integers named after Leonardo of Pisa, known as Fibonacci, who introduced the sequence to the Western world with his 1202 book Liber Abaci. The sequence has since become an essential concept in mathematics and computer science due to its frequent appearance in various natural phenomena and its applications in fields such as computational algorithms, financial markets, and theoretical physics.

Definition

A Fibonacci number is defined by the recurrence relation:

\( F(n) = F(n-1) + F(n-2) \)

with seed values:

\( F(0) = 0, \quad F(1) = 1 \)

This means the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the two preceding ones.

Sequence

The first few Fibonacci numbers are:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

Properties

Closed-form expression

The Fibonacci numbers can be expressed using Binet's formula:

\( F(n) = \fracTemplate:\phi^n - (1-\phi)^n{{\sqrt{5}}} \)

where \( \phi \) (phi) is the Golden Ratio, approximately equal to 1.618033988749895.

Divisibility

Fibonacci numbers exhibit a rich variety of divisibility properties. For instance, every third Fibonacci number is even, and more generally, \( F(n) \) is divisible by \( F(k) \) if and only if \( n \) is divisible by \( k \).

Golden Ratio

As \( n \) increases, the ratio \( \fracTemplate:F(n+1)Template:F(n) \) converges to the golden ratio \( \phi \). This property is often cited as one of the reasons Fibonacci numbers appear frequently in theories of aesthetics and design.

Applications

Fibonacci numbers appear in a variety of applications:

Nature

The branching patterns in trees, the arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of an artichoke, and the arrangement of a pine cone all display Fibonacci numbers.

Computer Algorithms

Many algorithms, especially those involving iterative and recursive solutions like quicksort and dynamic programming, employ Fibonacci numbers. They also appear in algorithms for computing greatest common divisors and for optimization in certain scenarios.

Financial Markets

The Fibonacci retracement technique is popular in technical analysis for predicting the future behavior of financial markets.

Generalizations

The Fibonacci sequence can be generalized to start with any two numbers. The resulting sequence, called the generalized Fibonacci sequence, retains similar properties to the original.

See Also

• Golden Spiral
• Pisano Period
• Fibonacci Prime

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