Fibonacci Calculator

Starting with 1, 1, each subsequent number is the sum of the previous two.

This creates the famous sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...

As Fibonacci numbers grow, the ratio between consecutive terms approaches the golden ratio φ ≈ 1.618.

This ratio appears throughout nature, art, and architecture.

A Fibonacci number is any term in the sequence where each value equals the sum of the two before it, beginning 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. The rule is F(n) = F(n-1) + F(n-2), with the two seed values F(1) = F(2) = 1.

Named after the 13th-century Italian mathematician Leonardo of Pisa, known as Fibonacci, the sequence was popularized in his 1202 book Liber Abaci through a rabbit-population puzzle.

As Wolfram MathWorld notes, it is one of the most studied integer sequences in mathematics, catalogued as sequence A000045 in the OEIS.

To find the nth Fibonacci number directly without listing every prior term, use Binet's formula: F(n) = (φⁿ − ψⁿ) / √5, where φ = (1 + √5)/2 ≈ 1.6180339887 is the golden ratio and ψ = (1 − √5)/2 ≈ −0.6180339887.

Because |ψ| is less than 1, ψⁿ shrinks toward zero, so F(n) is simply the nearest integer to φⁿ/√5. For example, φ¹⁰/√5 ≈ 55.004, which rounds to F(10) = 55.

NIST's Digital Library of Mathematical Functions (DLMF) documents this closed form alongside related Lucas-number identities.

The 10th Fibonacci number is 55 when the sequence is indexed starting at F(1) = 1. Counting the terms in order gives 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, so the tenth entry is 55.

Indexing matters: some references, including parts of the OEIS listing, start at F(0) = 0, which shifts every position by one and would make the tenth listed value 34.

This calculator uses the common 1-based convention F(1) = F(2) = 1. Always confirm whether a source counts from F(0) or F(1) before comparing answers.

A positive integer N is a Fibonacci number if and only if 5N² + 4 or 5N² − 4 is a perfect square. This test comes from the algebraic structure behind Binet's formula and lets you verify membership without generating the whole sequence.

Take N = 13: 5 × 13² − 4 = 5 × 169 − 4 = 841 = 29², a perfect square, so 13 is Fibonacci.

By contrast, N = 20 fails both checks (5 × 400 + 4 = 2004 and 5 × 400 − 4 = 1996 are not squares), so 20 is not a Fibonacci number.

Khan Academy covers the perfect-square reasoning that underpins this identity.

The sum of the first n Fibonacci numbers equals F(n+2) − 1, a tidy identity that avoids adding terms one by one. For the first 10 numbers, 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143, which matches F(12) − 1 = 144 − 1 = 143.

Two related identities are also worth knowing:

• the sum of the first n odd-indexed terms equals F(2n)
• the sum of squares F(1)² + … + F(n)² equals F(n) × F(n+1)

Wolfram MathWorld lists these summation formulas among the sequence's classic closed-form results.

The ratio of consecutive Fibonacci numbers converges to the golden ratio φ ≈ 1.6180339887 as the terms grow. Early ratios oscillate around φ: 55/34 ≈ 1.6176 and 89/55 ≈ 1.6182, tightening with each step.

This happens because Binet's formula makes φⁿ the dominant term. According to Encyclopaedia Britannica, φ satisfies the equation φ² = φ + 1, so φ ≈ 1.618 and its reciprocal 1/φ ≈ 0.618 differ by exactly 1.

The same constant appears in the pentagon's diagonals and in logarithmic spirals, which is why Fibonacci ratios recur across geometry and design.

Fibonacci numbers appear in far more than textbooks.

• In nature, sunflower seed heads, pinecone scales, and nautilus shells arrange along Fibonacci-related spirals that pack efficiently, a phenomenon documented by Encyclopaedia Britannica.
• In computer science, they benchmark recursion and dynamic programming — the same way a factorial calculator does for the other classic recursive definition — and appear in Fibonacci heaps and search algorithms.
• In finance, traders use Fibonacci retracement levels (23.6%, 38.2%, 61.8%) derived from ratios of the sequence.
• Agile software teams even use a Fibonacci-style scale (1, 2, 3, 5, 8, 13) for story-point estimation because the widening gaps reflect growing uncertainty in larger tasks.

Several common errors trip people up:

• The most frequent error is index confusion: mixing the F(0) = 0 convention with the F(1) = 1 convention shifts every term by one position and produces answers that are off by a step.
• A second pitfall is trusting Binet's formula for very large n on a standard calculator, where floating-point rounding can produce the wrong nearest integer; iterative addition or big-integer arithmetic stays exact.
• Others incorrectly assume every large number is Fibonacci without applying the 5N² ± 4 perfect-square test, or forget that 1 appears twice at the start.
• Finally, remember that no negative counting index gives a term below F(1) in this tool.

Fibonacci numbers belong to a broader family of second-order recurrences that share the rule term = previous + one-before-previous but use different starting values.

The Lucas numbers begin 2, 1, 3, 4, 7, 11, 18 and are catalogued as OEIS sequence A000032; they satisfy L(n) = φⁿ + ψⁿ and are closely tied to Fibonacci through identities such as F(2n) = F(n) × L(n). The Pell numbers (1, 2, 5, 12, 29) follow a different multiplier.

Studying these relatives, as Wolfram MathWorld details, clarifies why the golden ratio governs the growth rate of the whole family.