Fibonacci Sequence
Explore the Fibonacci sequence 0, 1, 1, 2, 3, 5, 8, 13 …, in which each term is the sum of the two before it. Look up any term F(n) for n up to 100,000, worked out exactly with whole-number (BigInt) arithmetic (F(100,000) has 20,899 digits), and download the first 1,000 terms with the Lucas numbers as CSV. Chart the ratio of each term to the one before and watch it close in on the golden ratio phi, for the Fibonacci numbers, the Lucas numbers or two starting numbers of your own. Print the classic tiling of squares whose sides are the Fibonacci numbers, with a quarter circle in each square, as a PDF, a 300 DPI PNG or an SVG. Nothing is uploaded.
100% private — everything runs in your browser. Everything runs in your browser; nothing is uploaded.
Print at 100% (Actual size). Real size prints on one sheet when the drawing fits; the check bar in the footer lets you confirm the scale. The SVG holds the drawing at the printed size. The CSV holds F(0) onwards with its digit count and the Lucas number (and your own sequence when you set one).
Fibonacci sequence
Exact terms
Look
Page
Runs in your browser with our own code: nothing you type is uploaded or saved. Every term is an exact whole number (BigInt), added up from the two before it, and every square, arc and ratio is drawn from those same terms. You may use what you make here, including commercially, with no credit needed.
How to explore the Fibonacci sequence
- Pick a preset, or choose the square tiling or the ratio chart.
- Set the number of squares (2 to 24), or the number of terms and the sequence for the chart.
- Type n to look up F(n), and choose how many terms go in the CSV.
- Choose the colours, arcs, labels and Fitted to one page or Real size.
- Download the PDF and print at 100% (Actual size), or the PNG, SVG or CSV.
What you can use it for
- Looking up any Fibonacci number exactly
- Lists of terms for spreadsheets and code tests
- Lessons on the golden ratio and convergence
- Square-tiling posters and line art
- Plotter and laser art from the SVG
- Trying sequences with your own starting numbers
The 16 presets
| Preset | Shows | Look-up |
|---|---|---|
| Squares 1 to 55 with arcs | 10 squares, largest 55 | F(100): 21 digits |
| Squares 1 to 21, labelled | 8 squares, largest 21 | F(100): 21 digits |
| Squares 1 to 144 | 12 squares, largest 144 | F(100): 21 digits |
| Squares 1 to 987 (16 squares) | 16 squares, largest 987 | F(100): 21 digits |
| Squares only, no arcs | 10 squares, largest 55 | F(100): 21 digits |
| Arcs only, no fill | 12 squares, largest 144 | F(100): 21 digits |
| Squares 1 to 89, warm colours | 11 squares, largest 89 | F(100): 21 digits |
| Squares 1 to 34, browns | 9 squares, largest 34 | F(100): 21 digits |
| Ratios F(n + 1) / F(n) approach phi | Fibonacci, 30 ratios | F(100): 21 digits |
| Ratios, first 12 terms | Fibonacci, 12 ratios | F(100): 21 digits |
| Lucas number ratios | Lucas, 30 ratios | F(100): 21 digits |
| F(1,000): exact, 209 digits | 10 squares, largest 55 | F(1,000): 209 digits |
| F(10,000): exact, 2,090 digits | 12 squares, largest 144 | F(10,000): 2,090 digits |
| F(100,000): exact, 20,899 digits | 14 squares, largest 377 | F(100,000): 20,899 digits |
| Your own start: 3, 7, 10, 17 … ratios | start 3, 7, 25 ratios | F(100): 21 digits |
| Real size: 1 = 2 mm, squares to 55 | 10 squares, largest 55 | F(100): 21 digits |
Every value is worked out by the tool.
The first 31 terms
| n | F(n) | Lucas L(n) | F(n + 1) / F(n) |
|---|---|---|---|
| 0 | 0 | 2 | – |
| 1 | 1 | 1 | 1 |
| 2 | 1 | 3 | 2 |
| 3 | 2 | 4 | 1.5 |
| 4 | 3 | 7 | 1.6666666667 |
| 5 | 5 | 11 | 1.6 |
| 6 | 8 | 18 | 1.625 |
| 7 | 13 | 29 | 1.6153846154 |
| 8 | 21 | 47 | 1.619047619 |
| 9 | 34 | 76 | 1.6176470588 |
| 10 | 55 | 123 | 1.6181818182 |
| 11 | 89 | 199 | 1.6179775281 |
| 12 | 144 | 322 | 1.6180555556 |
| 13 | 233 | 521 | 1.6180257511 |
| 14 | 377 | 843 | 1.6180371353 |
| 15 | 610 | 1,364 | 1.6180327869 |
| 16 | 987 | 2,207 | 1.6180344478 |
| 17 | 1,597 | 3,571 | 1.6180338134 |
| 18 | 2,584 | 5,778 | 1.6180340557 |
| 19 | 4,181 | 9,349 | 1.6180339632 |
| 20 | 6,765 | 15,127 | 1.6180339985 |
| 21 | 10,946 | 24,476 | 1.618033985 |
| 22 | 17,711 | 39,603 | 1.6180339902 |
| 23 | 28,657 | 64,079 | 1.6180339882 |
| 24 | 46,368 | 103,682 | 1.618033989 |
| 25 | 75,025 | 167,761 | 1.6180339887 |
| 26 | 121,393 | 271,443 | 1.6180339888 |
| 27 | 196,418 | 439,204 | 1.6180339887 |
| 28 | 317,811 | 710,647 | 1.6180339888 |
| 29 | 514,229 | 1,149,851 | 1.6180339887 |
| 30 | 832,040 | 1,860,498 | 1.6180339888 |
Big terms
| n | Digits in F(n) | F(n) |
|---|---|---|
| 10 | 2 | 55 |
| 100 | 21 | 354,224,848,179,261,915,075 |
| 1,000 | 209 | 4346655768…6849228875 |
| 10,000 | 2,090 | 3364476487…9947366875 |
| 100,000 | 20,899 | 2597406934…3428746875 |
Exact, by fast doubling; long values show the first and last digits.
Binet’s formula: φ^n / √5 rounds to F(n)
| n | F(n) | φ^n / √5 | Rounds to F(n) |
|---|---|---|---|
| 5 | 5 | 4.959675 | yes |
| 10 | 55 | 55.003636 | yes |
| 20 | 6,765 | 6765.00003 | yes |
| 30 | 832,040 | 832040 | yes |
| 40 | 102,334,155 | 102334155 | yes |
| 50 | 12,586,269,025 | 12586269025.000019 | yes |
| 60 | 1,548,008,755,920 | 1548008755920.00293 | yes |
| 70 | 190,392,490,709,135 | 190392490709135.4375 | yes |
Sums of terms and of squares
| n | F(1) + … + F(n) | Equals | F(1)² + … + F(n)² | Equals |
|---|---|---|---|---|
| 5 | 12 | F(7) − 1 | 40 | F(5) × F(6) |
| 10 | 143 | F(12) − 1 | 4,895 | F(10) × F(11) |
| 20 | 17,710 | F(22) − 1 | 74,049,690 | F(20) × F(21) |
| 50 | 32,951,280,098 | F(52) − 1 | 256,319,508,074,468,182,850 | F(50) × F(51) |
The square tiling
| Squares | Largest side | Rectangle | Long side / short side |
|---|---|---|---|
| 3 | 2 | 2 × 3 | 1.5 |
| 5 | 5 | 5 × 8 | 1.6 |
| 8 | 21 | 21 × 34 | 1.61904762 |
| 10 | 55 | 55 × 89 | 1.61818182 |
| 12 | 144 | 144 × 233 | 1.61805556 |
| 16 | 987 | 987 × 1,597 | 1.61803445 |
| 20 | 6,765 | 6,765 × 10,946 | 1.618034 |
| 24 | 46,368 | 46,368 × 75,025 | 1.61803399 |
The rectangle’s proportions approach φ as squares are added.
What prints
| Download | What is in it |
|---|---|
| The square tiling or the ratio chart fitted to one page or at real size, vector squares, arcs and labels, with the sizes and a check bar | |
| PNG 300 DPI | The same sheet as a picture |
| SVG | The drawing at the printed size: squares as rectangles, arcs as paths, labels as text |
| CSV | F(n), digits, L(n) (and your own sequence) for n from 0 |
For a cutter that needs DXF, use SVG to DXF; to work with φ in a layout, use the Golden Ratio Calculator.
Which tool to use
| You want | Use |
|---|---|
| Exact Fibonacci terms, ratios and the square tiling | Fibonacci Sequence (this page) |
| A spiral template to lay over a photo or layout | Golden Spiral |
| Golden-ratio proportions for a design | Golden Ratio Calculator, Golden Rectangle |
| Fibonacci numbers inside Pascal’s triangle | Pascal’s Triangle |
Other tools: the Golden Ratio Calculator splits lengths by φ, the Logo Grid Generator builds golden-ratio circle grids and the Spiral Art Generator draws other spirals.
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Frequently asked questions
What is the Fibonacci sequence?
The sequence that starts 0, 1 and then adds the two previous terms to get the next: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 … We write F(0) = 0, F(1) = 1 and F(n) = F(n − 1) + F(n − 2).
Who was Fibonacci?
Leonardo of Pisa, known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci. It had been described much earlier in Indian mathematics, as early as 200 BC in work by Pingala on patterns in Sanskrit poetry (Wikipedia, Fibonacci sequence, read 13 November 2026).
How big can n be?
Up to 100,000. Every term is exact: the tool uses BigInt whole numbers and the fast-doubling formulas F(2k) = F(k) (2 F(k + 1) − F(k)) and F(2k + 1) = F(k)² + F(k + 1)², which our tests check against adding the terms one by one up to F(10,000). F(100,000) has 20,899 digits; the box under the drawing holds every digit.
What is the golden ratio phi?
φ = (1 + √5) / 2 = 1.618033988750… The ratio of each Fibonacci number to the one before, 1/1, 2/1, 3/2, 5/3, 8/5 …, gets closer and closer to φ, landing alternately below and above it. From n = 40 on, the ratio is within 0.000000000000001 of φ (our tests check this).
What is Binet’s formula?
A closed form for the terms: F(n) = (φ^n − (−1/φ)^n) / √5. Because the second part shrinks quickly, F(n) is simply φ^n / √5 rounded to the nearest whole number. The table below checks this up to n = 70; beyond that, ordinary floating-point numbers are not precise enough, which is why the tool uses exact whole numbers instead.
What are the Lucas numbers?
The same rule with a different start: 2, 1, 3, 4, 7, 11, 18 … Each Lucas number is the sum of the two Fibonacci numbers either side of the same place, L(n) = F(n − 1) + F(n + 1), which our tests check. Their ratios approach φ as well.
Can I use my own starting numbers?
Yes: choose Your own two starting numbers and type a(0) and a(1), up to 30 digits each. Any start (other than two zeros) whose terms do not hit 0 gives ratios that approach φ too; the CSV adds your sequence as a column.
How does the square tiling work?
Start with two 1 × 1 squares side by side, then keep adding a square on the long side of the rectangle so far: 2, then 3, 5, 8 … Each new square’s side is the next Fibonacci number, and the squares always make a rectangle F(N) × F(N + 1). The quarter circle in each square joins the next one, making a spiral of arcs. Our tests check every side, that no squares overlap and that the arcs join.
Some sums worth knowing
The first n Fibonacci numbers add up to F(n + 2) − 1, and their squares add up to F(n) × F(n + 1), which is the area of the square tiling. The table below checks both for several n.
What is in the CSV?
For each n from 0: F(n), its number of digits, the Lucas number L(n), and your own sequence if you set one. Up to 1,000 terms.
Can it print at real size?
Yes, when the drawing fits on one sheet: choose Real size and set the size. Every printed cell, square and line is within 0.2 mm of its true place in our tests, on US Letter and A4, as long as you print at 100% (Actual size); the check bar in the footer lets you confirm the scale. Bigger drawings print fitted to the page.
Is it free?
Yes. There is no sign-up, no watermark and no limit.
Is anything uploaded?
No. Every number is worked out by JavaScript in your browser. Nothing you type is sent anywhere or saved when you close the page.