Fibonacci Sequence Generator
Fibonacci numbers from any two starting values: the first 1 to 1000 terms, the exact nth term up to n = 10,000, and the golden ratio estimate. How it works
Use the Fibonacci Sequence Generator
About the Fibonacci Sequence Generator
The first 20 Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1,597, 2,584 and 4,181, and F(50) = 12,586,269,025. Each term is the sum of the two before it, with F(0) = 0 and F(1) = 1; the generator also takes any other two whole numbers as starting values and shows the exact nth term, up to n = 10,000.
Fibonacci numbers F(n) from F(0) = 0 and F(1) = 1
| n | F(n) | Digits |
|---|---|---|
| F(10) | 55 | 2 |
| F(20) | 6,765 | 4 |
| F(30) | 832,040 | 6 |
| F(40) | 102,334,155 | 9 |
| F(50) | 12,586,269,025 | 11 |
| F(60) | 1,548,008,755,920 | 13 |
| F(70) | 190,392,490,709,135 | 15 |
| F(80) | 23,416,728,348,467,685 | 17 |
| F(90) | 2,880,067,194,370,816,120 | 19 |
| F(100) | 354,224,848,179,261,915,075 | 21 |
| F(200) | 280,571,172,992,510,140,037,611,932,413,038,677,189,525 | 42 |
The result shows these exact values with commas between digit groups. F(1000) has 209 digits and F(10000) has 2,090.
The golden ratio estimate by number of terms
| Terms listed | Last two terms | Ratio | Difference from φ |
|---|---|---|---|
| 5 | 3 ÷ 2 | 1.5000000000 | 0.118033988750 |
| 6 | 5 ÷ 3 | 1.6666666667 | 0.048632677917 |
| 8 | 13 ÷ 8 | 1.6250000000 | 0.006966011250 |
| 10 | 34 ÷ 21 | 1.6190476190 | 0.001013630298 |
| 12 | 89 ÷ 55 | 1.6181818182 | 0.000147829432 |
| 15 | 377 ÷ 233 | 1.6180257511 | 0.000008237677 |
| 20 | 4,181 ÷ 2,584 | 1.6180340557 | 0.000000066978 |
| 30 | 514,229 ÷ 317,811 | 1.6180339888 | 0.000000000004 |
The ratio of the last two listed terms, F(terms−1) ÷ F(terms−2), as the page prints it to 10 decimals, with its distance from φ = 1.6180339887 to 12 decimals. It does not depend on n.
The first 8 terms from other starting values
| A = F(0) | B = F(1) | First 8 terms |
|---|---|---|
| 0 | 1 | 0, 1, 1, 2, 3, 5, 8, 13 |
| 2 | 1 | 2, 1, 3, 4, 7, 11, 18, 29 |
| 1 | 3 | 1, 3, 4, 7, 11, 18, 29, 47 |
| 5 | 5 | 5, 5, 10, 15, 25, 40, 65, 105 |
| -1 | 1 | -1, 1, 0, 1, 1, 2, 3, 5 |
A = 2 and B = 1 is the Lucas sequence.
How to use this tool
- Enter Starting value A (F(0)) and Starting value B (F(1)). The defaults 0 and 1 give the usual sequence.
- Set Number of terms for the list and Position n for the single value F(n).
- Read F(n), the list from F(0) to F(terms−1), and the golden ratio estimate with its difference from φ. Copy puts the list on the clipboard.
Frequently asked questions
- What is the 10th Fibonacci number?
- F(10) = 55 when F(0) = 0 and F(1) = 1, the numbering this page uses. If you count the first number in the list as the first term, the 10th term is F(9) = 34: the list reads 0, 1, 1, 2, 3, 5, 8, 13, 21, 34.
- How do I find the nth Fibonacci number?
- Enter n in Position n and read the F(n) result. F(100) = 354,224,848,179,261,915,075 has 21 digits, F(1000) has 209 digits and F(10000) has 2,090 digits, all exact, because the page adds whole numbers as BigInt values. A standard JavaScript number stops being exact at F(79).
- How close is the ratio of Fibonacci numbers to the golden ratio?
- 55 ÷ 34 (11 terms) is 1.6176470588, 0.000386929926 below φ = 1.6180339887, and 46,368 ÷ 28,657 (25 terms) is 1.6180339882, 0.000000000545 below it. By 40 terms the ratio matches φ in all 10 decimals the page prints. The ratios alternate around φ: 3 ÷ 2 = 1.5 is below it and 5 ÷ 3 = 1.6666666667 is above it.
- Can I start with numbers other than 0 and 1?
- Yes. Any two whole numbers from −999,999,999 to 999,999,999 work as A and B. A = 2 and B = 1 give the Lucas numbers 2, 1, 3, 4, 7, 11, 18, 29, and A = 5 and B = 5 give 5, 5, 10, 15, 25, 40. The ratio of neighbors still heads for φ: the Lucas start gives 1.6180339218 after 20 terms.
- Why does the golden ratio show a dash?
- The estimate divides the last listed term by the one before it, so it needs at least two terms and is undefined when that earlier term is 0. With the defaults and 2 terms the earlier term is F(0) = 0, and A = 0 with B = 0 gives all zeros. The page then shows a dash and says no ratio can be computed.
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