Generate Fibonacci sequences, find the Nth Fibonacci number, check if any number is Fibonacci, and watch golden ratio convergence. Includes a Fibonacci spiral canvas visual, Lucas numbers, and a golden ratio convergence table.
Three modes: Generate Sequence (produces the first N Fibonacci numbers), Find Nth Term (computes F(n) directly), or Check Number (tests whether a given number is a Fibonacci number). The golden ratio φ ≈ 1.6180339887 is shown alongside each pair of consecutive terms.
For sequence mode, enter how many terms you want (up to 100). For Nth term, enter n. For the checker, enter any positive integer. The tool also shows Lucas numbers (2, 1, 3, 4, 7, 11…) — the sibling sequence with the same recurrence but different starting values.
The golden ratio convergence table shows F(n)/F(n-1) approaching φ as n increases. The Fibonacci spiral canvas draws the classic rectangular spiral. Copy the sequence as comma-separated values or download as text.
The Fibonacci sequence is a series where each number is the sum of the two preceding ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89… Starting with F(0)=0 and F(1)=1, the rule is F(n) = F(n−1) + F(n−2). It was introduced to Western mathematics by Leonardo of Pisa (Fibonacci) in 1202 to model rabbit population growth, though the sequence appeared in Indian mathematics centuries earlier in the context of Sanskrit prosody (syllable patterns).
The golden ratio φ (phi) ≈ 1.6180339887 is an irrational number satisfying φ² = φ + 1. As n → ∞, the ratio of consecutive Fibonacci numbers F(n)/F(n−1) converges to φ. For example: F(10)/F(9) = 55/34 ≈ 1.6176, F(20)/F(19) = 6765/4181 ≈ 1.6180. This relationship is given by Binet's formula: F(n) = (φⁿ − ψⁿ)/√5 where ψ = (1−√5)/2 ≈ −0.618. The golden ratio appears in art, architecture, biology (phyllotaxis), and financial analysis (Fibonacci retracements in trading).
A positive integer N is a Fibonacci number if and only if one or both of (5N² + 4) and (5N² − 4) is a perfect square. This is a known mathematical property. For example, for N=13: 5×169+4=849 (not a perfect square), 5×169−4=841=29² (perfect square) → 13 is Fibonacci. This calculator applies this test instantly for any number you enter.
Lucas numbers follow the same recurrence as Fibonacci (L(n) = L(n−1) + L(n−2)) but start with L(0)=2 and L(1)=1, giving: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76… Lucas numbers share many properties with Fibonacci numbers and are named after French mathematician Édouard Lucas (1842–1891), who also coined the name "Fibonacci numbers". The ratio of consecutive Lucas numbers also converges to φ. Lucas numbers appear in the analysis of Fibonacci-related algorithms and number theory.
Fibonacci numbers appear throughout nature in biological growth patterns. Sunflower seed spirals: typically 34 clockwise and 55 counterclockwise spirals (consecutive Fibonacci). Pinecone scales: 8 and 13 spirals. Pineapple rows: 8, 13, and 21. Romanesco broccoli floret spirals. Nautilus shell growth (approximately logarithmic spiral related to golden ratio). Flower petals: many species have 3, 5, 8, or 13 petals. Tree branch patterns follow Fibonacci numbers. This is because Fibonacci-structured growth allows for optimal packing and space efficiency.
This tool computes Fibonacci numbers using BigInt (arbitrary precision integers in modern browsers) for exact results. JavaScript's standard floating-point Number type loses precision beyond F(52) ≈ 3.3 × 10¹⁰. With BigInt, the tool correctly computes F(100) = 354,224,848,179,261,915,075 and beyond. For practical display purposes the tool shows up to F(200) in the sequence generator mode, with digits formatted for readability.
