What is the Fibonacci sequence?

A sequence where each term is the sum of the two before it, beginning 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. It was introduced to European mathematics by Leonardo of Pisa in 1202 as a puzzle about breeding rabbits, though it appears in Indian prosody several centuries earlier. The rabbit problem is artificial, but the sequence turned out to describe a great deal else, largely because the rule — add the last two — is the simplest possible way for a quantity to grow out of its own recent history.

How is the golden ratio connected?

Divide any term by the one before it and the result approaches φ = (1 + √5) ÷ 2 = 1.618033988749895. The approach alternates — one ratio above φ, the next below — and the error shrinks by roughly a factor of φ² each step. F(10) ÷ F(9) is 1.617647059, already close; by the fortieth term the ratio matches φ to every digit a double can hold. This is not a coincidence about Fibonacci specifically: any sequence built by adding the last two terms converges to the same ratio, whatever two numbers it starts from.

How do I get the nth term without listing them all?

Two ways. Binet's formula gives F(n) = (φⁿ − ψⁿ) ÷ √5 directly, where ψ is (1 − √5) ÷ 2; it is exact in principle but relies on irrational numbers, so in floating point it starts rounding wrong for large n. The method used here is fast doubling, which computes F(2k) and F(2k+1) from F(k) and F(k+1) and so reaches the thousandth term in about ten steps rather than a thousand — and it stays in whole numbers throughout, so nothing is ever approximated.

Why does this need big integers?

Because ordinary computer numbers stop being exact at 9,007,199,254,740,991, and the sequence passes that point at the 79th term. F(78) is 8,944,394,323,791,464, which fits; F(79) is 14,472,334,024,676,221, which does not. A page using ordinary arithmetic does not fail loudly there — it returns a number that looks entirely plausible and is wrong in its last few digits, and stays wrong for every term after. This calculator uses exact big-integer arithmetic throughout, which is why F(500) can be shown to all 105 of its digits.

How can I tell whether a number is a Fibonacci number?

There is an exact test: N is a Fibonacci number if and only if 5N² + 4 or 5N² − 4 is a perfect square. Nothing else passes it, and nothing in the sequence fails it. Take 100: neither 50,004 nor 49,996 is a perfect square, so 100 is not in the sequence — it falls between F(11) = 89 and F(12) = 144. The test is exact rather than approximate, which matters for large numbers where comparing against φⁿ ÷ √5 would run into rounding.

What is the Zeckendorf representation?

Every positive whole number can be written as a sum of Fibonacci numbers, no two of them next to each other in the sequence — and there is exactly one way to do it. For 100 that is 89 + 8 + 3, using indices 11, 6 and 4, which are pairwise non-adjacent. The greedy approach of repeatedly taking the largest Fibonacci number that fits always produces this unique representation, which is a stronger guarantee than greedy algorithms usually offer.

Do Fibonacci numbers really appear in nature?

Some of the famous claims are true and some are not, and it is worth separating them. The spiral counts in sunflower heads, pinecones and pineapples genuinely are Fibonacci numbers, and there is a real mechanism: placing each new growth at roughly 137.5° from the last — the golden angle — packs seeds most efficiently, and that packing produces Fibonacci spiral counts. Branching patterns and petal counts often follow too. The nautilus shell, by contrast, is a logarithmic spiral but not a golden one, and claims that the Parthenon or the Mona Lisa were designed around φ are retrofitted rather than documented.

What are Lucas numbers?

The same rule with different seeds: 2 and 1 instead of 0 and 1, giving 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123. They interleave with Fibonacci in a tidy way — L(n) = F(n−1) + F(n+1), so L(10) = 34 + 89 = 123 — and their ratios converge to φ just as quickly, since any add-the-last-two sequence does. They matter practically because the Lucas–Lehmer test, which is how the largest known primes are found, is built on a closely related sequence.

Is the Fibonacci spiral the same as the golden spiral?

No, and the difference is worth knowing. The picture on this page is built from quarter circles, one per square, each with a constant radius — so it is a sequence of circular arcs joined end to end, and its curvature jumps at every join. A true golden spiral is a smooth logarithmic curve whose radius grows continuously by a factor of φ every quarter turn. The two look nearly identical because the Fibonacci ratios approach φ so quickly, and the arcs approximate the smooth curve better with every square, but they are different objects.

Fibonacci Calculator

The rule is the simplest in mathematics — every term is the sum of the two before it, starting from 0 and 1 — and almost everything interesting about the sequence follows from that one line. This page works in both directions. Give it a position and it returns that term exactly, along with its neigh...

BY POSITION

INDEX n

WORTH TRYING

F(10)

RATIO TO F(9)

1.617647

2 digits · previous 34 · next 89 · φ is 1.618034, so this is off by 3.9e-4

Digits

Lucas L(n)

123

Sum of F(1)…F(n)

143

Ratio to previous

1.617647059

Exact in a double?

yes

Index n

AROUND F(10)

F(n)

DIGITS

144

233

STEPS

Every term is the sum of the two before it, starting from F(0) = 0 and F(1) = 1

F(10) = F(9) + F(8) = 34 + 21

2 digits, small enough to hold exactly in ordinary arithmetic

F(10) = 55

GOLDEN SPIRAL

The squares build the sequence. Each new square takes the whole long side of what is already there, so its side is the sum of the two before it — which is the Fibonacci rule stated in geometry rather than arithmetic. The rectangle is always F(k) by F(k+1), and the quarter arcs join end to end into the spiral. It approaches the golden spiral without ever being one: this is built from circular arcs, and the true golden spiral is a smooth logarithmic curve. Square 10 is picked out in gold.

THE RATIO CLOSING ON φ

n = 3

2.00000000

n = 5

1.66666667

n = 8

1.61538462

n = 12

1.61797753

n = 20

1.61803396

n = 40

1.61803399

Bar length is the distance from φ = 1.618033989, so shorter is closer. The ratio alternates above and below it and the error falls by roughly a factor of φ² each step, which is why n = 40 already matches φ to the last digit a double can hold.

Exact, not floating point. F(78) is the last term that fits exactly in ordinary double arithmetic; F(79) is 14,472,334,024,676,221, which is past the limit. Everything here is computed with exact big integers, so F(500) is right to the final digit rather than rounded. Indices run to 1,000.

Live simulation · drawn to scale · updates as you type

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HOW TO USE

1
In TERM mode, enter a position from 0 to 1000. You get that Fibonacci number in full, its digit count, the terms either side, the Lucas number at the same index, the running total of everything up to it, and the ratio to the previous term with its distance from φ.
2
Try 78 and then 79. The first is the last term that fits exactly in ordinary computer arithmetic and the second is not, which is the single most common source of wrong answers on other Fibonacci pages. Both are exact here.
3
Switch to IS IT ONE? and type any whole number up to thirty digits. The 5N² ± 4 test settles it immediately, and if the number is not in the sequence you get the two terms it falls between plus its Zeckendorf representation — the one and only way to write it as a sum of non-consecutive Fibonacci numbers.
4
Watch the spiral build, or drag the scrubber a square at a time. Each square takes the full long side of the rectangle so far, so its side is the sum of the two before it, and the rectangle is always F(k) by F(k+1).

WORKED EXAMPLE

What is the 10th Fibonacci number? Start from F(0) = 0 and F(1) = 1 and apply the rule ten times: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. So F(10) = 55. The terms either side are F(9) = 34 and F(11) = 89. Dividing 55 by 34 gives 1.617647059, which is already within 0.0004 of φ = 1.618033988749895 — and it is below φ, because the ratios alternate. Three things follow from the factorisation of the sequence rather than from more counting. The running total F(1) + … + F(10) is 143, which is F(12) − 1 = 144 − 1, and that identity holds for every n. The Lucas number at the same index is L(10) = F(9) + F(11) = 34 + 89 = 123. And the sum of the squares up to F(10) is F(10) × F(11) = 55 × 89 = 4,895. Now the other direction. Is 100 a Fibonacci number? The test asks whether 5 × 100² + 4 = 50,004 or 5 × 100² − 4 = 49,996 is a perfect square. Neither is, so 100 is not in the sequence — it falls between F(11) = 89 and F(12) = 144. It can still be written using Fibonacci numbers, though, in exactly one way with no two of them adjacent: 100 = 89 + 8 + 3, at indices 11, 6 and 4.

THE FORMULAS

The ruleF(n) = F(n−1) + F(n−2)

The seedsF(0) = 0, F(1) = 1

Golden ratioφ = (1 + √5) ÷ 2

Binet's formulaF(n) = (φⁿ − ψⁿ) ÷ √5

Ratio limitF(n) ÷ F(n−1) → φ

Sum of the first nF(1) + … + F(n) = F(n+2) − 1

Sum of squaresF(1)² + … + F(n)² = F(n)·F(n+1)

Cassini identityF(n)² − F(n−1)F(n+1) = (−1)ⁿ⁻¹

Greatest common divisorgcd(F(m), F(n)) = F(gcd(m,n))

Lucas numbersL(n) = F(n−1) + F(n+1)

Fibonacci test5N² ± 4 is a perfect square

Zeckendorfa unique sum of non-consecutive terms

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Last updated: July 29, 2026 · Formula verified · Exact big-integer arithmetic, so terms past F(78) are right to the final digit · Eagle-eyed accuracy for every calculation.