Lucas Number Calculator
Lucas number with Fibonacci relation.
What it does
Lucas numbers shadow Fibonacci with a head start: 2, 1 instead of 0, 1, then the same add the pair rule. Every Lucas splits into neighboring Fibonacci numbers.
The tool above takes index n. It returns L(n) with the Fibonacci relation and neighbors.
The default n = 10 gives 123 as F(9) + F(11), or 34 + 89. The n = 5 case gives 11, and n = 0 gives the seed 2.
Sequence references tabulate Lucas from the same recurrence that Fibonacci texts teach. The neighbor identity bridges both families in one line.
Golden ratios haunt both: Lucas over Fibonacci nears phi powers while closed forms round. Integer stepping keeps this page exact through 75.
How to use
- Type n from 0 to 75.
- Read L(n) first.
- Read the Fibonacci relation beside it.
- Read neighbors last.
- Compare with Fibonacci pages.
- Respect the 75 cap.
Formula
- n
- Index 0 to 75.
- L(n)
- nth Lucas number.
- F
- Neighbor Fibonacci numbers.
OEIS A000204 tabulates Lucas numbers 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123 with the Fibonacci neighbor identity. LibreTexts recurrence chapters mirror the same second order build. Integers stay exact through n = 75 in doubles.
Worked example
Example: Lucas 10
Default: n = 10.
- Index: n = 10.
- Build: 2, 1, 3, 4, 7, 11, ... to 123.
- Value: L(10) = 123.
- Relation: F(9) + F(11) = 34 + 89.
- Neighbors: F(9)=34 with F(11)=89.
- Tool shows 123, L(10) = F(9) + F(11), F(9)=34, F(11)=89.
Reference table
| n | L(n) | Relation |
|---|---|---|
| 0 | 2 | Seed |
| 1 | 1 | Seed |
| 5 | 11 | 3 + 8 |
| 10 | 123 | 34 + 89 |
| 12 | 322 | 89 + 233 |
Limits
- n 0 to 75 for exactness.
- Single index per run.
- Integers only.
- Float ceilings past 75.
Common mistakes
- Seeding 0, 1 like Fibonacci.
- Off by one neighbor picks.
- Expecting exact past 75.
- Reading Lucas as Fibonacci.
- Skipping the relation check.
Try it yourself
Quick checks. Answers sit tucked under each question.
n = 0?
L(0) = 2, the seed, with F(0)=0 and F(1)=1.
n = 5?
L(5) = 11 with F(4)=3 and F(6)=8.
n = 99?
Past 75 floats round: enter 0 to 75.
Questions
Lucas vs Fibonacci?
Same sum rule as Fibonacci from 2, 1 instead of 0, 1. Every Lucas equals neighbor Fibonacci sums.
Relation?
L(n) = F(n-1) + F(n+1). L(10) = 34 + 89 = 123.
Used where?
Sunflowers, primes testing, and Diophantine puzzles share the sequence.
Golden ratio link?
Yes, near phi^n with Lucas exactness beating floats past 75.
Whole sequence?
No, this page runs one index. Ranges need scripting loops.
Large n exact?
Yes, past 75 floats round. Exact big integers need libraries.