Need a specific Lucas number or want to see the series? This Lucas series calculator generates any Lucas number up to the 100,000th position and shows the full sequence up to the 2,500th term.
Lucas Series Calculator
Using the Lucas calculator
Enter the Position (n) of the Lucas number you want. The calculator instantly shows that number and (optionally) the entire series from 0 to n.
Indexing starts at 0: the 0th Lucas number is 2, the 1st is 1, the 2nd is 3, and so on.
What is the Lucas series?
The Lucas series follows the same rule as the Fibonacci sequence – each term is the sum of the previous two – but with different starting values:
With starting values:
The first 20 Lucas numbers are:
Lucas and Fibonacci numbers are intertwined. There's a beautiful identity connecting them:
Every Lucas number is the sum of the Fibonacci numbers immediately before and after it at that position. For example:
L₅ = 11andF₄ + F₆ = 3 + 8 = 11L₁₀ = 123andF₉ + F₁₁ = 34 + 89 = 123
And vice versa:
The golden ratio connection
Like Fibonacci, the ratio of consecutive Lucas numbers converges to the golden ratio (φ):
The calculator shows this ratio so you can watch it approach φ as n increases.
Closed-form formula
There's a closed-form expression for Lucas numbers similar to Binet's formula for Fibonacci:
Where φ = (1 + √5) / 2 and ψ = (1 - √5) / 2.
This is actually simpler than the Fibonacci formula (no division by √5). Since |ψ| < 1, for large n you can approximate:
Example: the 50th Lucas number
The 50th Lucas number is:
That's over 28 billion – slightly larger than F₅₀ = 12,586,269,025. Lucas numbers grow at the same exponential rate as Fibonacci, just with larger values.
Who was Lucas?
François Édouard Anatole Lucas (1842-1891) was a French mathematician who studied the Fibonacci sequence extensively. He gave it the name "Fibonacci sequence" (after the medieval mathematician Leonardo of Pisa) and discovered many of its properties. The Lucas numbers bearing his name have similar properties and appear in related mathematical contexts.
