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Fibonacci

 Fibonacci[n]gives the Fibonacci number . Fibonacci[n, x]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 .
• Fibonacci can be evaluated to arbitrary numerical precision.
Compute Fibonacci numbers:
Compute Fibonacci numbers:
 Out[1]=
 Scope   (7)
Evaluate large Fibonacci numbers:
Fibonacci numbers of negative argument:
Non-integer arguments:
Complex arguments:
Series expansion at generic point:
Fibonacci polynomials:
General series expansion at infinity:
 Applications   (10)
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: