BUILT-IN MATHEMATICA SYMBOL

Fibonacci

Fibonacci[n]
gives the Fibonacci number

.

Fibonacci

gives the Fibonacci polynomial

.

MORE INFORMATION

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.

EXAMPLES

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Basic Examples (1)

Compute Fibonacci numbers:

In[1]:=

Out[1]=

Scope (7)

Evaluate large Fibonacci numbers:

Fibonacci numbers of negative argument:

Non-integer arguments:

Complex arguments:

Fibonacci threads element-wise over lists:

Series expansion at generic point:

TraditionalForm formatting:

Generalizations & Extensions (2)

Fibonacci polynomials:

General series expansion at infinity:

Applications (12)

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

:

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

:

Properties & Relations (10)

Expand in terms of elementary functions:

Limiting ratio:

Explicit recursive definition:

Explicit state-space recursive definition:

Closed-form 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:

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:

Fibonacci modulo n :

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

Fibonacci number:

Contours of vanishing real and imaginary parts of Fibonacci:

LogPlot of positive and negative Fibonacci numbers:

While the Fibonacci numbers are nondecreasing for non-negative arguments, the Fibonacci function possesses a single local minimum:

Since the generating function is rational these sums come out as rational numbers: