How this calculator works
Add the two preceding terms to produce the next term. Standard Fibonacci numbers start with F₀ = 0 and F₁ = 1; other integer seeds follow the same recurrence. The term calculation uses exact integers rather than the floating-point Binet formula. The displayed Binet coefficients are approximate explanatory values, and their closed form is not used to generate the exact output.
Formula
Fₙ₊₂ = Fₙ₊₁ + Fₙ. Binet form: Fₙ = Aφⁿ + Bψⁿ, where φ = (1+sqrt(5))/2, ψ = (1-sqrt(5))/2, A = (F₁-ψF₀)/sqrt(5), B = F₀-A.
Worked example
With seeds 0 and 1, the sequence begins 0, 1, 1, 2, 3, 5 and F₁₀ = 55. With seeds 2 and 1, it begins 2, 1, 3, 4, 7, 11, so F₅ = 11.
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Frequently asked questions
Is the first term index zero or one?
This calculator uses index zero for the first seed and index one for the second seed.
Can I enter negative seeds?
Yes. The recurrence still produces exact integers, although ratios and growth behaviour can differ from the standard sequence.
Why stop at index 5000?
The digits grow with the index. The bound keeps the exact result manageable, while the table shows only the first 25 terms.