Online Calculators

# Fibonacci Calculator

Fibonacci Number Generator

F15 = 610

$F_{n}=F_{n-1}+F_{n-2}$$F_{15}=F_{14}+F_{13}$
F15 = 377 + 233

F15 = 610

Solution Formula:
$F_{n}={\dfrac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}\sqrt{5}}}$$F_{15}=\dfrac{(1+\sqrt{5})^{15}-(1-\sqrt{5})^{15}}{2^{15}\sqrt{5}}$$F_{15} = \dfrac{ \phi^{15} - \psi^{15} }{ \sqrt{5}}$$F_{15}=\dfrac{(1.618..)^{15}-(-0.618..)^{15}}{\sqrt{5}}$
F15 = 610

## Calculator Use

With the Fibonacci calculator you can generate a list of Fibonacci numbers from start and end values of n. You can also calculate a single number in the Fibonacci Sequence, Fn, for any value of n up to n = ±500.

## Fibonacci Sequence

The Fibonacci Sequence is a set of numbers such that each number in the sequence is the sum of the two numbers that immediatly preceed it.

$F_{0} = 0,\quad F_{1} = F_{2} = 1,$

and

$F_{n}=F_{n-1}+F_{n-2}$

For example, calculating F4

$F_{4}=F_{4-1}+F_{4-2}$ $F_{4}=F_{3}+F_{2}$ $F_{4}=2+1$ $F_{4}=3$

The first 15 numbers in the sequence, from F0 to F14, are

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377

## Fibonacci Sequence Formula

The formula for the Fibonacci Sequence to calculate a single Fibonacci Number is:

$F_{n}={\dfrac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}\sqrt{5}}}$

or

Fn = ( (1 + √5)^n - (1 - √5)^n ) / (2^n × √5)

for positive and negative integers n.

A simplified equation to calculate a Fibonacci Number for only positive integers of n is:

$F_{n}=\left[{\dfrac{(1+\sqrt{5})^{n}}{2^{n}\sqrt{5}}}\right]$

or

Fn = [( (1 + √5)^n ) / (2^n × √5)]

where the brackets in [x] represent the nearest integer function. Simply put, this means to round up or down to the closest integer.

A more compact version of the formula used is:

$F_{n} = \dfrac{ \phi^{n} - \psi^{n} }{ \sqrt{5}}$

or

Fn = ( φ^n - ψ^n ) / √5

where φ, the Greek letter phi, is the Golden Ratio φ = (1 + √5) / 2 ≈ 1.618034... and ψ, the Greek letter psi, is ψ = (1 - √5) / 2 ≈ -0.618034...

Since it can be shown that ψ^n is small and gets even smaller as n gets larger, when only working with positive integers of n, the compact Fibonacci Number formula is true:

$F_{n} = \left[ \dfrac{\phi^n}{\sqrt{5}} \right] = \left[ {\dfrac{(1+\sqrt{5})^{n}}{2^{n}\sqrt{5}}}\right]$

where the brackets in [x] represent the nearest integer function as defined above.

### Negative Fibonacci Numbers

Unless stated otherwise, formulas above will hold for negative values of n however, it could be easier to find Fn and solve for F-n using the following equation.

$F_{-n}=(-1)^{n+1}F_{n}$

Putting it another way, when -n is odd, F-n = Fn and when -n is even, F-n = -Fn.

If you are generating a sequence of -n by hand and working toward negative infinity, you can restate the sequence equation above and use this as a starting point:

$F_{0} = 0,\quad F_{1} = F_{2} = 1,$

and

$F_{n}=F_{n+2}-F_{n+1}$

For example with n = -4 and referencing the table below

$F_{-4}=F_{-4+2}-F_{-4+1}$ $F_{-4}=F_{-2}-F_{-3}$ $F_{-4}=-1-2$ $F_{-4}=-3$

F-9  to  F9

n
Fn
-9
34
-8
-21
-7
13
-6
-8
-5
5
-4
-3
-3
2
-2
-1
-1
1
0
0
1
1
2
1
3
2
4
3
5
5
6
8
7
13
8
21
9
34

## References

Knuth, D. E., The Art of Computer Programming. Volume I. Fundamental Algorithms, Addison-Wesley, 1997, Boston, Massachusetts. pages 79-86

Chandra, Pravin and Weisstein, Eric W. "Fibonacci Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FibonacciNumber.html

Cite this content, page or calculator as:

Furey, Edward "Fibonacci Calculator" at https://www.calculatorsoup.com/calculators/discretemathematics/fibonacci-calculator.php from CalculatorSoup, https://www.calculatorsoup.com - Online Calculators