Fibonacci

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 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:

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 [more info]:

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: