Fibonacci & Lucas Sequence Calculator
Generate the Fibonacci, Lucas, and Tribonacci sequences and compute individual terms with exact big-integer arithmetic. Find the nth term (via fast doubling, O(log n)), list the first N terms, sum the first N terms, test whether a number is a Fibonacci number and find its index, and compare the exact value with Binet's closed-form approximation and the golden-ratio convergence. Everything runs locally in your browser.
First N terms
Is a given number Fibonacci?
The Fibonacci sequence is F(0)=0, F(1)=1, F(n)=F(n−1)+F(n−2): 0, 1, 1, 2, 3, 5, 8, 13… The Lucas numbers use the same recurrence but start L(0)=2, L(1)=1: 2, 1, 3, 4, 7, 11… Tribonacci sums the previous three terms from T(0)=0, T(1)=0, T(2)=1. The ratio F(n+1)/F(n) tends to the golden ratio φ = (1+√5)/2 ≈ 1.618. Binet's formula F(n) = (φⁿ − ψⁿ)/√5 gives a closed form, shown alongside the exact value. Fast doubling computes F(n) in O(log n) using F(2k)=F(k)(2F(k+1)−F(k)) and F(2k+1)=F(k+1)²+F(k)². All arithmetic uses BigInt, so values stay exact even for F(1000) (209 digits). Pairs with the Number Base Converter and Prime tools. Everything runs locally — nothing leaves your browser.