Need a specific Fibonacci number or want to see the sequence? This Fibonacci sequence calculator generates any Fibonacci number up to the 100,000th position and shows the full sequence up to the 2,500th term.
Fibonacci Sequence Calculator
Using the Fibonacci calculator
Enter the Position (n) of the Fibonacci number you want. The calculator instantly shows that number and (optionally) the entire sequence from 0 to n.
Indexing starts at 0: the 0th Fibonacci number is 0, the 1st is 1, the 2nd is 1, and so on.
What is the Fibonacci sequence?
The Fibonacci sequence is defined by adding the two previous numbers to get the next, starting with 0 and 1:
With starting values:
The first 20 Fibonacci numbers are:
The golden ratio connection
As you go further in the sequence, the ratio of consecutive Fibonacci numbers converges to the golden ratio (φ):
For example:
8 / 5 = 1.60013 / 8 = 1.62521 / 13 = 1.615...144 / 89 = 1.61798...10946 / 6765 = 1.618033...
The calculator shows this ratio below the result so you can watch it approach φ as n increases.
Binet's formula
There's also a closed-form expression for finding any Fibonacci number directly:
Where φ = (1 + √5) / 2 and ψ = (1 - √5) / 2.
Since |ψ| < 1, the ψ term shrinks rapidly, so you can approximate:
However, for large n, floating-point precision becomes an issue. The calculator uses BigInt arithmetic for exact results.
Example: the 50th Fibonacci number
The 50th Fibonacci number is:
That's already over 12 billion. By the 100th term, you're at 354,224,848,179,261,915,075 (over 354 quintillion). The sequence grows fast.
Fibonacci in nature and design
Fibonacci numbers appear throughout nature:
- Flower petals: Lilies have 3, buttercups have 5, delphiniums have 8, marigolds have 13
- Seed spirals: Sunflower heads typically show 34 and 55 spirals, or 55 and 89
- Pinecones: Usually 8 spirals one way, 13 the other
- Branching: Tree branches and leaf arrangements often follow Fibonacci patterns
The connection to the golden ratio explains this: it's the most "irrational" number (hardest to approximate with fractions), which means seeds/petals arranged this way pack most efficiently without overlapping.
