Lucas Numbers Calculator
Compute the nth Lucas number and display the Lucas sequence up to that term.
About Lucas Numbers Calculator
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
Worked example
With the default values below, Lucas Numbers Calculator returns L(n) of 123. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Term Number (n)accepts 0–1,000
- 10
Results
- L(n)
- 123
- Digits
- 3
- Sequence (first terms)
- 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123
Step by step
- 1Recurrence
- 2L(10)= 123
Assumptions and limits
- The result is derived only from the single input listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
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.