The Fibonacci sequence
Each term is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and on. The list below and every ratio in it are computed here, from the definition.
Updated on October 7, 2026
The sequence, and the ratio between terms
The third column divides each term by the one before it. The ratio closes in on the golden ratio φ = 1.6180339887…, alternating above and below as it goes.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1,597, 2,584, 4,181, 6,765, …
| n | F(n) — Term | Ratio to the previous term |
|---|---|---|
| 1 | 1 | — |
| 2 | 1 | 1.000000000 |
| 3 | 2 | 2.000000000 |
| 4 | 3 | 1.500000000 |
| 5 | 5 | 1.666666667 |
| 6 | 8 | 1.600000000 |
| 7 | 13 | 1.625000000 |
| 8 | 21 | 1.615384615 |
| 9 | 34 | 1.619047619 |
| 10 | 55 | 1.617647059 |
| 11 | 89 | 1.618181818 |
| 12 | 144 | 1.617977528 |
| 13 | 233 | 1.618055556 |
| 14 | 377 | 1.618025751 |
| 15 | 610 | 1.618037135 |
| 16 | 987 | 1.618032787 |
| 17 | 1,597 | 1.618034448 |
| 18 | 2,584 | 1.618033813 |
| 19 | 4,181 | 1.618034056 |
| 20 | 6,765 | 1.618033963 |
| 21 | 10,946 | 1.618033999 |
| 22 | 17,711 | 1.618033985 |
| 23 | 28,657 | 1.618033990 |
| 24 | 46,368 | 1.618033988 |
| 25 | 75,025 | 1.618033989 |
| 26 | 121,393 | 1.618033989 |
| 27 | 196,418 | 1.618033989 |
| 28 | 317,811 | 1.618033989 |
| 29 | 514,229 | 1.618033989 |
| 30 | 832,040 | 1.618033989 |
| 31 | 1,346,269 | 1.618033989 |
| 32 | 2,178,309 | 1.618033989 |
| 33 | 3,524,578 | 1.618033989 |
| 34 | 5,702,887 | 1.618033989 |
| 35 | 9,227,465 | 1.618033989 |
| 36 | 14,930,352 | 1.618033989 |
| 37 | 24,157,817 | 1.618033989 |
| 38 | 39,088,169 | 1.618033989 |
| 39 | 63,245,986 | 1.618033989 |
| 40 | 102,334,155 | 1.618033989 |
φ = 1.6180339887
Source: Cálculo próprio a partir dos dados citados nesta página
The closed form, and how to get a term without the others
Binet’s formula gives F(n) directly: F(n) = (φⁿ − (−1/φ)ⁿ)/√5, where φ = (1+√5)/2. It looks like it should produce irrational numbers, and the two irrational halves cancel exactly, leaving an integer every time. In floating-point arithmetic it matches the true value up to F(70) or so and then drifts, which is why the list here is built by addition and not by the formula.
Two practical consequences of the same fact. F(n) is about φⁿ/√5, so the terms grow geometrically — F(100) already has 21 digits. And the ratio F(n+1)/F(n) converges to φ fast enough that F(17)/F(16) = 1597/987 is already right to five decimals.
Source: OEIS — The On-Line Encyclopedia of Integer Sequences
Where it really shows up, and where it does not
The honest examples come from growth that repeats a rule. Sunflower heads, pine cones and pineapples have spiral counts that are consecutive Fibonacci numbers, because leaves placed at an angle of 360°/φ² pack most tightly, and plants that do this catch more light. The sequence also counts real combinatorial things: the number of ways to tile a 1×n strip with squares and dominoes is exactly F(n+1).
The famous claims that do not hold up: the Parthenon, the Mona Lisa, the nautilus shell and credit cards are not built on φ. The nautilus is a logarithmic spiral, but its growth factor is nowhere near φ. Fibonacci’s own rabbit problem, from the Liber Abaci of 1202, is not biology either — it assumes immortal rabbits that breed on a fixed schedule. The sequence is genuinely beautiful without the embroidery.
Source: OEIS — The On-Line Encyclopedia of Integer Sequences
How to read this page
How to read the numbers here
Every value on "The Fibonacci sequence" comes from the source named beside its section, never from an estimate of our own with no origin.
Where the data comes from
Each section of "The Fibonacci sequence" names its official source; the verification date sits at the top of the page.
Linked to the rest of the module
"The Fibonacci sequence" is one of this module’s reference pages; the glossary below defines the technical terms used here.
More about “The Fibonacci sequence”
How close the ratio gets to φ
F(30) is already 832,040, and F(31)/F(30) works out to 1.618033989 — an approximation of φ correct to nine decimal places, one term away. F(100), from Binet’s formula on "The Fibonacci sequence", already runs to 21 digits.
Words on this page
Standard uncertainty
The margin attached to a measured value, usually written in parentheses on the last digits. A constant that is "exact by definition" has zero standard uncertainty; a measured one, like the gravitational constant G, always carries some.
Standard definition by BIPM — Bureau International des Poids et Mesures (SI) · Standard uncertainty
Mole
The SI counting unit for atoms, molecules or other entities: one mole is exactly 6.02214076 × 10²³ of them, by the definition fixed in 2019. It is what ties a mass in grams to a number of particles.
Standard definition by BIPM — Bureau International des Poids et Mesures (SI) · Mole
Atomic number
The number of protons in an atom’s nucleus, written Z. It is this number, not the mass, that defines which element an atom is: every atom with 8 protons is oxygen, whether it has 8 or 18 neutrons.
Standard definition by IUPAC · Atomic number
Atomic weight
The average weight of an element’s atoms, weighted by how abundant each isotope is in nature, in atomic mass units. For 14 elements CIAAW publishes a range instead of a single value, because that abundance varies a little from sample to sample.
Standard definition by CIAAW — Commission on Isotopic Abundances and Atomic Weights · Atomic weight