Lucas Numbers Calculator
Compute the nth Lucas number and display the Lucas sequence up to that term.
Inputs
Which Lucas number to compute (L(0)=2, L(1)=1, …). Max 1000.
L(n)
123
Digits
3
Sequence (first terms)
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123
Step by step
Recurrence
L(n) = L(n−1) + L(n−2), L(0)=2, L(1)=1
L(10)
= 123
How it works
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 instead of 0 and 1. The sequence begins 2, 1, 3, 4, 7, 11, 18, 29, … Like Fibonacci numbers, the ratio of consecutive Lucas numbers converges to the golden ratio φ. Lucas numbers satisfy the identity L(n) = F(n−1) + F(n+1).
Formulas
Lucas recurrence
L(n) = L(n−1) + L(n−2), L(0) = 2, L(1) = 1
- n
- Term index (0-based)
Relation to Fibonacci
L(n) = F(n−1) + F(n+1)
- F(k)
- kth Fibonacci number
- L(n)
- nth Lucas number
Frequently Asked Questions
How are Lucas numbers different from Fibonacci numbers?
They share the same recurrence rule (each term is the sum of the two preceding terms) but differ in initial conditions: Fibonacci starts 0, 1 while Lucas starts 2, 1.
What are Lucas numbers used for?
They appear in primality testing (Lucas-Lehmer test for Mersenne primes), combinatorial identities, and the theory of continued fractions. They also satisfy many identities that mirror those of Fibonacci numbers.