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Lucas Number Calculator

Find any Lucas number Lₙ.

Result

123

How it is calculated

The Lucas sequence starts L0 = 2, L1 = 1 and continues with Ln = Ln−1 + Ln−2. The tool simply adds the previous two terms n − 1 times, so it uses repeated addition, not a closed formula. n is cut to a whole number and must be between 0 and 80; outside that range nothing is returned. Values near the top of the range are very large, but every term up to n = 80 is computed exactly, so all digits shown are correct. Counting starts at 0, so L0 is the first term.

Worked example

For n = 10, write out the terms: L0 = 2, L1 = 1, then 3, 4, 7, 11, 18, 29, 47, 76 and finally 47 + 76 = 123. Each term is the sum of the two before it. L10 is 123.

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Common mistakes

  • Starting the count at 1. Here n = 0 gives 2 and n = 1 gives 1, so the tenth printed term (n = 9) is 76, not 123.
  • Mixing it up with Fibonacci, which starts 0, 1. The rule is the same but the seeds differ, so L10 = 123 while F10 = 55.
  • Entering n above 80 or a negative number and expecting a value. Both give no result.
  • Typing a decimal such as 10.7 and expecting rounding. The fraction is dropped, so it is treated as 10.

Sources

About this calculator

This calculator finds the nth term of the Lucas sequence, defined by the same recurrence as Fibonacci (Lₙ = Lₙ₋₁ + Lₙ₋₂) but with different starting values L₀=2, L₁=1, producing 2, 1, 3, 4, 7, 11, 18, 29, … Lucas numbers share the same golden-ratio-based closed form (Binet-like formula) as Fibonacci and satisfy several identities linking the two sequences, such as L(n) = F(n−1) + F(n+1).

Math students and recreational mathematicians use it to explore number theory identities and the golden ratio, and it appears in some algorithm analyses (like worst-case Euclidean algorithm inputs) and in a small number of primality tests (Lucas-Lehmer variants).

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Comments (1)

  • Editorial team

    Lucas numbers share the Fibonacci rule but start from 2 and 1. A neat check: each Lucas number equals the Fibonacci number just before it plus the one just after it, so L10 = 123 is F9 + F11 = 34 + 89. The tool accepts n from 0 to 80, and every term in that range is computed exactly, even the very large ones near 80. Why do you need Lucas numbers, puzzles or study?

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