Fibonacci Sequence Calculator
Generate the Fibonacci sequence and explore its mathematical properties.
Sequence Parameters
Custom Starting Values
Fibonacci Sequence
01123581321345589144233377610987159725844181
Lucas Numbers
213471118294776123199322521843
Related sequence starting with 2, 1
F(10)
55
Golden Ratio (phi)
1.6180339887
Sum of Sequence
10,945
Even Numbers
7
Odd Numbers
13
Ratio Convergence to phi
| n | F(n)/F(n-1) | |ratio - phi| |
|---|---|---|
| 2 | 1.0000000000 | 6.1803e-1 |
| 3 | 2.0000000000 | 3.8197e-1 |
| 4 | 1.5000000000 | 1.1803e-1 |
| 5 | 1.6666666667 | 4.8633e-2 |
| 6 | 1.6000000000 | 1.8034e-2 |
| 7 | 1.6250000000 | 6.9660e-3 |
| 8 | 1.6153846154 | 2.6494e-3 |
| 9 | 1.6190476190 | 1.0136e-3 |
| 10 | 1.6176470588 | 3.8693e-4 |
| 11 | 1.6181818182 | 1.4783e-4 |
| 12 | 1.6179775281 | 5.6461e-5 |
| 13 | 1.6180555556 | 2.1567e-5 |
| 14 | 1.6180257511 | 8.2377e-6 |
Binet's Formula
F(n) = (phi^n - psi^n) / sqrt(5)
phi = (1 + sqrt(5))/2, psi = (1 - sqrt(5))/2
Properties
- F(n) = F(n-1) + F(n-2)
- Sum of first n terms = F(n+2) - 1
- Every 3rd number is divisible by 2
- Every 4th number is divisible by 3
- F(n)^2 + F(n+1)^2 = F(2n+1)