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Fibonacci Recurrence with Generalized Seeds

Evaluate a Fibonacci-type recurrence at an integer index, with editable first two terms and negative-index support.

  • Formula & worked example
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

With F₀ = 0 and F₁ = 1, the tenth term is 55.

Fₙ = Fₙ₋₁ + Fₙ₋₂, with the specified first two terms and its equivalent closed form.

Worked example

Enter these known values and leave the other values blank.

Integer index n
10
Sequence term Fₙ
55

Assumptions and limitations

  • The retained index range is −250 through 250, inclusive, and the index must be an integer.
  • The captured closed form uses 64-digit arithmetic. Standard Fibonacci values in this range have accurate rounded integer displays; the internal value can contain a tiny closed-form rounding residual.
  • Editable real-valued seeds define a generalized recurrence. Very large seed magnitudes can exceed exact-integer precision; the result is a 64-significant-digit approximation.
  • Negative terms extend the same recurrence backwards. For standard seeds, F₋ₙ = (−1)ⁿ⁺¹Fₙ.
  • Arithmetic uses 64 significant decimal digits, with separate rounding for displayed values.

Common questions

Can I start from terms other than zero and one?

Enter different F₀ and F₁ values to define a new recurrence with the same sum-of-previous-two rule.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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