Lucas Number Calculator
The Lucas sequence is defined by L(0) = 2, L(1) = 1, and L(n) = L(n-1) + L(n-2) for every subsequent term. It was studied by the French mathematician Edouard Lucas in the 19th century and shares the same additive recurrence as the Fibonacci sequence, but with different starting values. Enter any term index n from 0 to 50 and this calculator returns L(n), displays the complete sequence from L(0) through L(n) in a table, and shows the ratio L(n)/L(n-1) converging toward the golden ratio phi = (1+sqrt(5))/2 = 1.6180339887. The relationship between Lucas and Fibonacci numbers is deep: L(n) = F(n-1) + F(n+1), and the closed-form expression is L(n) = phi^n + psi^n where psi = (1-sqrt(5))/2. Lucas numbers are used in the Lucas-Lehmer primality test for Mersenne primes, in combinatorics, and in number theory. For n above about 25, the values grow rapidly into large integers.
Lucas sequence table (first n+1 terms)
| n | L(n) | Ratio L(n)/L(n-1) |
|---|
Lucas number formula
L(0) = 2, L(1) = 1
L(n) = L(n-1) + L(n-2) for n ≥ 2
Closed form: L(n) = phi^n + psi^n
where phi = (1 + sqrt(5)) / 2 = 1.6180339887...
and psi = (1 - sqrt(5)) / 2 = -0.6180339887...
Lucas numbers: frequently asked questions
What are Lucas numbers?
Lucas numbers are a sequence similar to the Fibonacci sequence, defined by L(0)=2, L(1)=1, and L(n)=L(n-1)+L(n-2) for n greater than 1. The first several terms are 2, 1, 3, 4, 7, 11, 18, 29, 47, 76. Named after French mathematician Edouard Lucas (1842-1891).
How are Lucas numbers related to Fibonacci numbers?
Lucas numbers and Fibonacci numbers are closely related. The identity L(n) = F(n-1) + F(n+1) connects them, where F is the Fibonacci sequence. Also, L(n) = phi^n + psi^n where phi = (1+sqrt(5))/2 (the golden ratio) and psi = (1-sqrt(5))/2, while F(n) = (phi^n - psi^n)/sqrt(5).
Do Lucas numbers approach the golden ratio?
Yes. The ratio L(n)/L(n-1) approaches the golden ratio phi = (1+sqrt(5))/2 approximately 1.6180339887 as n increases, just as consecutive Fibonacci numbers do. For example, L(10)/L(9) = 123/76 approximately 1.6184, already within 0.03% of the golden ratio.
What are Lucas numbers used for?
Lucas numbers appear in primality testing (Lucas-Lehmer test for Mersenne primes), combinatorics, and number theory. They also appear in certain tilings and in the study of phyllotaxis (leaf arrangement in plants), and in computer algorithms related to the Fibonacci heap data structure.
Is there a closed form for L(n)?
Yes. L(n) = phi^n + psi^n, where phi = (1+sqrt(5))/2 and psi = (1-sqrt(5))/2. Since |psi| is less than 1, psi^n approaches zero, so L(n) is the nearest integer to phi^n for all n. This is Binet's formula for Lucas numbers.
Sources
- Lucas sequences and primality testing: Wolfram MathWorld, Lucas Sequence.
- Golden ratio and Fibonacci relations: NIST SP 330 Mathematical Constants.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.