Fibonacci Calculator
Calculate any term in the Fibonacci sequence and see the full list, the running sum, the golden-ratio convergence, and a growth chart. Go beyond the standard sequence: use negative indices or set your own starting values to explore any Fibonacci-like series.
Formula
Worked example
For n = 10: start with 0, 1, then repeatedly add the last two terms: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. So F(10) = 55. The sum of all eleven terms equals F(12) - 1 = 144 - 1 = 143. The ratio 55/34 = 1.617647, already within 0.02% of the golden ratio.
What the Fibonacci sequence is
The Fibonacci sequence starts with 0 and 1, and every term after that is the sum of the two preceding terms: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and onward without end. This calculator is 0-indexed by default, meaning F(0) = 0 and F(1) = 1, matching the standard mathematical convention used by the OEIS and most references. The sequence is built iteratively rather than recursively, keeping it fast and exact even for large positions. For positions above around 79 the numbers grow too big for ordinary floating-point arithmetic, so this tool uses arbitrary-precision integer math to return every digit exactly.
Negative indices and the extended sequence
The Fibonacci sequence can be extended backward into negative positions using the backward recurrence F(n-2) = F(n) - F(n-1). The result follows a neat pattern: for the standard sequence, F(-n) = F(n) times (-1)^(n+1), so negative-index terms alternate in sign and have the same magnitude as their positive counterparts. For example, F(-1) = 1, F(-2) = -1, F(-3) = 2, F(-4) = -3, and so on. Enable "Include negative-index terms" to see this reflected extension in the sequence table.
Custom starting values and generalizations
Any pair of starting values produces a valid Fibonacci-like sequence that follows the same addition rule. The Lucas sequence, starting with L(0) = 2 and L(1) = 1, is the most famous example: 2, 1, 3, 4, 7, 11, 18, .... Despite the different seeds, the ratio of consecutive terms in any such sequence converges to the same golden ratio. Use the "Custom starting values" mode to explore these generalizations and verify this convergence yourself.
Binet's formula and the golden ratio
Binet's closed-form formula computes F(n) directly without iteration: F(n) = (phi^n - psi^n) / sqrt(5), where phi = (1 + sqrt(5)) / 2 is the golden ratio and psi = (1 - sqrt(5)) / 2 is its conjugate. This formula is mathematically exact but limited in practice because floating-point exponentiation loses precision for large n (above about n = 75). The calculator shows the Binet result alongside the exact iterative value for n up to 75 so you can compare them. The deeper insight is that every Fibonacci number is simply the nearest integer to phi^n / sqrt(5), confirming how tightly the golden ratio governs the sequence's growth.
The sum identity and other properties
The sum of all Fibonacci numbers from F(0) to F(n) always equals F(n+2) minus 1. This identity makes it trivial to compute running totals once you know the sequence, and the calculator displays it in the steps panel with your actual values substituted in. Other notable identities include: F(2n) = F(n) times (2 F(n+1) - F(n)), and Cassini's identity F(n+1) F(n-1) - F(n)^2 = (-1)^n, which is useful in proofs. These properties make Fibonacci numbers appear throughout number theory, combinatorics, and algorithm analysis.
Fibonacci numbers F(0) to F(20) with digit counts
| n | F(n) | Digits | F(n)/F(n-1) |
|---|---|---|---|
| 0 | 0 | 1 | - |
| 1 | 1 | 1 | - |
| 2 | 1 | 1 | 1.000000 |
| 3 | 2 | 1 | 2.000000 |
| 4 | 3 | 1 | 1.500000 |
| 5 | 5 | 1 | 1.666667 |
| 6 | 8 | 1 | 1.600000 |
| 7 | 13 | 2 | 1.625000 |
| 8 | 21 | 2 | 1.615385 |
| 9 | 34 | 2 | 1.619048 |
| 10 | 55 | 2 | 1.617647 |
| 12 | 144 | 3 | 1.618182 |
| 15 | 610 | 3 | 1.618056 |
| 20 | 6,765 | 4 | 1.618034 |
Standard sequence: F(0)=0, F(1)=1. Each term is the sum of the previous two.
Frequently asked questions
Does the sequence start at 0 or 1?
This calculator is 0-indexed by default: F(0) = 0, F(1) = 1, F(2) = 1, F(3) = 2, and so on. If you need the 1-indexed convention where the first term is 1, select "Alternative (F(1)=1, F(2)=1)" from the sequence type menu.
What are negative Fibonacci numbers?
The recurrence F(n) = F(n-1) + F(n-2) can be rearranged to F(n-2) = F(n) - F(n-1), which extends the sequence into negative indices. For the standard sequence the pattern is F(-n) = F(n) times (-1)^(n+1), so the magnitudes mirror the positive sequence but alternate in sign: F(-1) = 1, F(-2) = -1, F(-3) = 2, F(-4) = -3.
How large a Fibonacci number can this calculator handle?
Up to n = 1000 with exact results. Fibonacci numbers use arbitrary-precision integer arithmetic internally, so even F(1000), which has 209 digits, comes back with every digit correct rather than as a rounded approximation.
What is Binet's formula?
Binet's formula gives F(n) = (phi^n - psi^n) / sqrt(5), where phi = (1 + sqrt(5)) / 2 is the golden ratio. It is a closed-form expression, meaning you can jump straight to any term without computing all the ones before it. It is exact in principle but loses floating-point precision around n = 75, which is why the calculator shows the exact iterative result alongside the Binet approximation.
What happens if I use custom starting values?
The same F(n) = F(n-1) + F(n-2) rule applies, just starting from your chosen seeds. The ratio of consecutive terms still converges to the golden ratio no matter which seeds you pick, which is a fundamental property of any sequence obeying this recurrence.
What is the sum of Fibonacci numbers up to F(n)?
The sum of F(0) through F(n) always equals F(n+2) minus 1. For example, 0+1+1+2+3+5+8+13+21+34+55 = 143 = F(12) - 1 = 144 - 1. This identity holds for the standard sequence regardless of how large n is.