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gives the Fibonacci number .
gives the Fibonacci polynomial .
  • 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.
Compute Fibonacci numbers:
Compute Fibonacci numbers:
Click for copyable input
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:
Fibonacci polynomials:
General series expansion at infinity:
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 :
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:
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:
Fibonacci numbers modulo 10:
Fibonacci modulo n :
Count the number of 1, 2, ..., 9, 0 digits in the 1,000,000^(th) 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:
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