Fibonacci Sequence
Generate the Fibonacci sequence
About This Calculator
The Fibonacci sequence is one of the most famous number patterns in mathematics, where each term is the sum of the two preceding terms. It appears in nature (flower petals, spiral shells, branching plants) and is closely related to the golden ratio.
Formula
F(n) = F(n-1) + F(n-2)
F(0) = 0, F(1) = 1
Closed form (Binet): F(n) = (phi^n - psi^n) / sqrt(5), where phi = (1+sqrt(5))/2 ≈ 1.618
Example Calculation
First 10 Fibonacci terms
- F(0)=0, F(1)=1
- F(2)=1, F(3)=2, F(4)=3, F(5)=5
- F(6)=8, F(7)=13, F(8)=21, F(9)=34
0, 1, 1, 2, 3, 5, 8, 13, 21, 34
Fibonacci Numbers F(1) through F(20)
| n | F(n) | n | F(n) |
|---|---|---|---|
| 1 | 1 | 11 | 89 |
| 2 | 1 | 12 | 144 |
| 3 | 2 | 13 | 233 |
| 4 | 3 | 14 | 377 |
| 5 | 5 | 15 | 610 |
| 6 | 8 | 16 | 987 |
| 7 | 13 | 17 | 1597 |
| 8 | 21 | 18 | 2584 |
| 9 | 34 | 19 | 4181 |
| 10 | 55 | 20 | 6765 |
Frequently Asked Questions
What is the golden ratio and how does it relate to Fibonacci?
The golden ratio phi ≈ 1.618 is the limit of F(n+1)/F(n) as n grows. Adjacent Fibonacci numbers get closer and closer to this ratio: 5/3=1.667, 8/5=1.6, 13/8=1.625, converging toward phi.
Where does the Fibonacci sequence appear in nature?
Sunflower seed spirals, pinecone scales, pineapple bracts, and the arrangement of leaves on a stem all follow Fibonacci numbers. The pattern optimizes packing and resource use.
What is a Fibonacci spiral?
A Fibonacci spiral is drawn by connecting the corners of squares whose side lengths are successive Fibonacci numbers. It approximates the golden spiral and appears frequently in shells, galaxies, and art.
Can Fibonacci numbers be negative?
Yes, the sequence can be extended backwards using F(n-2) = F(n) - F(n-1), giving: ..., -8, 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, ... These are called negafibonacci numbers.