Lucas Number Calculator
Fibonacci’s lesser-known sibling sequence, with the identity that ties the two together verified for whichever term you pick.
Result
The Fibonacci-Lucas identity checks out: L(n) equals F(n−1) + F(n+1) exactly.
| n | L(n) |
|---|---|
| 0 | 2 |
| 1 | 1 |
| 2 | 3 |
| 3 | 4 |
| 4 | 7 |
| 5 | 11 |
| 6 | 18 |
| 7 | 29 |
| 8 | 47 |
| 9 | 76 |
| 10 | 123 |
| 11 | 199 |
| 12 | 322 |
Same rule, different starting point
It’s easy to assume the Fibonacci sequence is special because 0-then-1 is somehow a privileged starting pair, but the recurrence F(n) = F(n−1) + F(n−2) works from any two starting numbers. Start it from 2 and 1 instead, and you get the Lucas sequence — a completely legitimate integer sequence with its own identities, its own appearances in number theory (it's central to the classical Lucas–Lehmer primality test for Mersenne primes), and the same long-run connection to the golden ratio.
The identity this calculator checks — L(n) = F(n−1) + F(n+1) — is one of several that link the two sequences algebraically, a reminder that Fibonacci numbers aren’t an isolated curiosity but one member of a broader family defined by the same simple rule.
Frequently Asked Questions
What are the Lucas numbers?
The Lucas sequence uses the exact same recurrence as Fibonacci — each term is the sum of the two before it — but starts from L(0)=2 and L(1)=1 instead of 0 and 1. That gives 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123… Édouard Lucas, the 19th-century mathematician the sequence is named for, is also the person who gave the 'Fibonacci sequence' its modern name.
What is the Fibonacci–Lucas identity?
L(n) = F(n−1) + F(n+1) — every Lucas number can be written as the sum of the Fibonacci numbers on either side of the matching index. This calculator computes both sides independently (the Lucas recurrence, and the Fibonacci terms via the negafibonacci-extended recurrence) and shows that they agree.
What's the 'negafibonacci' extension used at n=0?
The identity needs F(n−1), which means F(−1) when n=0. Fibonacci numbers extend cleanly to negative indices using F(−n) = (−1)ⁿ⁺¹F(n), giving F(−1)=1, F(−2)=−1, F(−3)=2, and so on. With that extension, F(−1) + F(1) = 1 + 1 = 2 = L(0), so the identity holds even at the boundary.
Do Lucas numbers relate to the golden ratio the way Fibonacci numbers do?
Yes — the ratio of consecutive Lucas numbers also converges to φ as n grows, for the same reason it does for Fibonacci: both sequences satisfy the same recurrence, so both are dominated by the φⁿ term as n increases and the contribution from the ψⁿ term fades away.
Educational tool. Results are exact integers for every supported term (n ≤ 70), matching the same double-precision limit as the Fibonacci calculator.