The Fibonacci sequence — 0, 1, 1, 2, 3, 5, 8, 13, 21, ... — is one of the most famous patterns in mathematics. Each number is the sum of the two before it, and the deeper you go, the more the sequence reveals: a closed-form formula, a surprising link to the golden ratio, and appearances throughout nature. This calculator finds any term exactly, lists the sequence, and visualizes that golden-ratio convergence.
How the sequence is built
The rule could not be simpler: start with 0 and 1, then each new term is the sum of the previous two. That gives 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and on forever. Because every term depends only on the last two, you can compute F(n) by walking up from the bottom — which is exactly what this calculator does, using arbitrary-precision integers so nothing is ever rounded. The terms grow roughly geometrically: each is about 1.618 times the last, which is the first hint of the golden ratio hiding inside the sequence.
Binet's formula: a shortcut to any term
You do not have to loop to find F(n). Binet's formula gives it directly: 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. It looks like it should produce an irrational mess, yet it always lands on a whole number, because the irrational parts cancel. Since |psi| is less than 1, the psi^n term fades to nothing as n grows, so F(n) is simply the nearest integer to phi^n / sqrt(5). This calculator shows Binet's estimate next to the exact BigInt value so you can see how the two agree.
The golden ratio connection
Divide any Fibonacci number by the one before it — 5/3, 8/5, 13/8, 21/13 — and the answers home in on phi ≈ 1.6180339887, the golden ratio. The ratios alternate, overshooting then undershooting phi, with the gap shrinking fast. Phi is special in its own right: it is the only positive number where phi = 1 + 1/phi, and it shows up in pentagons, the geometry of spirals, and art and architecture that aim for balanced proportions. The Golden Ratio tab on this page charts the convergence directly against a reference line at phi.
Lucas numbers, negative indices, and nature
Keep the same recurrence but change the seeds and you get a whole family of sequences. Seeds 2 and 1 give the Lucas numbers (2, 1, 3, 4, 7, 11, 18, ...), which interlock with the Fibonacci numbers in elegant identities. The sequence even runs backwards: the negafibonacci rule F(-n) = (-1)^(n+1) F(n) extends it to negative indices with alternating signs. In the wild, Fibonacci numbers count the spirals in sunflower heads and pinecones and the arrangement of leaves around a stem — not because nature "knows" the sequence, but because packing seeds or leaves at the golden angle (derived from phi) is an efficient way to fill space.