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Lucas Number — Definition, Formula & Examples

A Lucas number is a term in the integer sequence 2, 1, 3, 4, 7, 11, 18, 29, ... where each term is the sum of the two preceding terms, starting with 2 and 1.

The Lucas sequence {Ln}\{L_n\} is defined by the recurrence relation Ln=Ln−1+Ln−2L_n = L_{n-1} + L_{n-2} for n≥2n \geq 2, with initial conditions L0=2L_0 = 2 and L1=1L_1 = 1. It shares the same recurrence as the Fibonacci sequence but differs in its starting values.

Key Formula

Ln=Ln−1+Ln−2,L0=2,  L1=1L_n = L_{n-1} + L_{n-2}, \quad L_0 = 2,\; L_1 = 1
Where:
  • LnL_n = The nth Lucas number
  • Ln−1L_{n-1} = The previous Lucas number
  • Ln−2L_{n-2} = The Lucas number two positions before

How It Works

To generate Lucas numbers, start with L0=2L_0 = 2 and L1=1L_1 = 1. Then add the two most recent terms to get the next one. For instance, L2=1+2=3L_2 = 1 + 2 = 3, L3=3+1=4L_3 = 3 + 1 = 4, L4=4+3=7L_4 = 4 + 3 = 7, and so on. The ratio of consecutive Lucas numbers approaches the golden ratio ϕ≈1.618\phi \approx 1.618, just as it does for Fibonacci numbers. In fact, there is a direct connection: Ln=Fn−1+Fn+1L_n = F_{n-1} + F_{n+1}, where FnF_n denotes the nnth Fibonacci number.

Worked Example

Problem: Find the 8th Lucas number, L_8.
List known values: Write out the sequence starting from the initial conditions.
L0=2,  L1=1,  L2=3,  L3=4,  L4=7,  L5=11,  L6=18,  L7=29L_0=2,\; L_1=1,\; L_2=3,\; L_3=4,\; L_4=7,\; L_5=11,\; L_6=18,\; L_7=29
Apply the recurrence: Add the two most recent terms to find L_8.
L8=L7+L6=29+18=47L_8 = L_7 + L_6 = 29 + 18 = 47
Answer: L8=47L_8 = 47

Why It Matters

Lucas numbers appear in number theory proofs involving Fibonacci identities and in primality testing algorithms. Understanding this sequence strengthens your ability to work with recurrence relations, a skill used throughout discrete mathematics and computer science.

Common Mistakes

Mistake: Confusing the starting values with Fibonacci's. Students often begin the Lucas sequence with 0 and 1 (or 1 and 1) instead of 2 and 1.
Correction: The Fibonacci sequence starts F0=0,F1=1F_0=0, F_1=1. The Lucas sequence starts L0=2,L1=1L_0=2, L_1=1. The recurrence rule is identical, but the initial values are different and produce an entirely different sequence.