# Fibonacci

Fibonacci[n]

gives the Fibonacci number .

Fibonacci[n,x]

gives the Fibonacci polynomial .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• The satisfy the recurrence relation with .
• For any complex value of n, the are given by the general formula , where is the golden ratio.
• The Fibonacci polynomial is the coefficient of in the expansion of .
• The Fibonacci polynomials satisfy the recurrence relation .
• FullSimplify and FunctionExpand include transformation rules for combinations of Fibonacci numbers with symbolic arguments when the arguments are specified to be integers using nIntegers.
• Fibonacci can be evaluated to arbitrary numerical precision.
• Fibonacci automatically threads over lists.
• Fibonacci can be used with Interval and CenteredInterval objects. »

# Examples

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## Basic Examples(6)

Compute Fibonacci numbers:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Series expansion at a singular point:

## Scope(42)

### Numerical Evaluation(5)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

Fibonacci can be used with Interval and CenteredInterval objects:

### Specific Values(6)

Values of Fibonacci at fixed points:

Fibonacci polynomial for symbolic n and x:

Values at zero:

Find the value of in which :

Compute the Fibonacci[7,x] polynomial:

Compute Fibonacci[1/2,x]:

### Visualization(5)

Plot the Fibonacci function:

Plot the Fibonacci polynomial for various orders:

Plot the real part of :

Plot the imaginary part of :

Plot as real parts of two parameters vary:

Types 2 and 3 of Fibonacci polynomial have different branch cut structures:

### Function Properties(14)

Fibonacci is defined for all real values:

Approximate function range of Fibonacci:

Fibonacci polynomial of an even order is odd:

Fibonacci polynomial of an odd order is even:

Fibonacci has the mirror property :

Fibonacci is an analytic function of x:

Fibonacci is neither non-decreasing nor non-increasingfor odd values:

Fibonacci is non-decreasing for even values:

Fibonacci is not injective for odd values:

Fibonacci is not surjective for odd values:

Fibonacci is non-negative for odd values:

Fibonacci has no singularities or discontinuities:

Fibonacci is convex for odd values:

### Differentiation(3)

First derivatives with respect to n:

First derivative with respect to x:

Higher derivatives with respect to n:

Plot the higher derivatives with respect to n:

Formula for the  derivative with respect to n:

### Integration(3)

Compute the indefinite integral using Integrate:

Definite integral:

More integrals:

### Series Expansions(4)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

General term in the series expansion using SeriesCoefficient:

Find the series expansion at Infinity:

Taylor expansion at a generic point:

### Function Identities and Simplifications(2)

The ordinary generating function of Fibonacci:

Recurrence relation:

## Generalizations & Extensions(2)

Fibonacci polynomials:

General series expansion at infinity:

## Applications(13)

Solve the Fibonacci recurrence equation:

Find ratios of successive Fibonacci numbers:

Compare with continued fractions:

Convergence to the golden ratio:

Fibonacci substitution system:

Fibonomial coefficients:

Calculate the number of ways to write an integer as a sum of Fibonacci numbers :

Plot the counts for the first hundred integers:

Lamé's theorem bounds the number of steps of the Euclidean algorithm for calculating :

Plot the maximal number of steps:

Find the first Fibonacci number above 1000000:

Plot the discrete inverse of Fibonacci numbers:

Plot of the absolute value of Fibonacci over the complex plane:

Find the number of factors of Fibonacci polynomials:

If divides , then divides :

This is a particular case of a more general identity :

The sequence of is periodic with respect to for a fixed natural number :

For , the period equals :

Build Zeckendorf's representation of a positive integer [MathWorld]:

Define Fibonacci multiplication for positive integers:

Fibonacci multiplication table:

Verify that the Fibonacci multiplication is associative:

## Properties & Relations(15)

### Fibonacci Numbers(13)

Expand in terms of elementary functions:

Limiting ratio:

Explicit recursive definition:

Explicit statespace recursive definition:

Closedform solution using MatrixPower:

Simplify expressions involving Fibonacci numbers:

Symbolic summation:

Generating function:

Fibonacci numbers as coefficients:

Express a fractional Fibonacci number as an algebraic number:

Fibonacci can be represented as a DifferenceRoot:

General term in the series expansion of Fibonacci:

The generating function for Fibonacci:

FindSequenceFunction can recognize the Fibonacci sequence:

The exponential generating function for Fibonacci:

### Fibonacci Polynomials(2)

Expand in terms of elementary functions:

Explicitly construct Fibonacci polynomials:

## Possible Issues(3)

Large arguments can give results too large to be computed explicitly: Results for integer arguments may not hold for non-integers:

Matrix power representation is valid only for integers:

## Neat Examples(8)

Fibonacci numbers modulo 10:

Count the number of 1, 2, ..., 9, 0 digits in the 1,000,000 Fibonacci number: