Negative Fibonacci Number Calculator
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Reviewed by Calculator Editorial Team
The Fibonacci sequence is a famous mathematical series where each number is the sum of the two preceding ones, starting from 0 and 1. However, the sequence can be extended to include negative numbers, creating the negative Fibonacci numbers. This calculator helps you compute these extended values and understand their properties.
What is Negative Fibonacci?
The standard Fibonacci sequence is defined by the recurrence relation:
Standard Fibonacci Sequence
Fn = Fn-1 + Fn-2
With initial conditions: F0 = 0, F1 = 1
The sequence extends to negative indices by using the same recurrence relation but with different initial conditions. The negative Fibonacci numbers are defined as:
Negative Fibonacci Sequence
F-n = (-1)n+1 × Fn
This extension creates a symmetric sequence that includes both positive and negative indices. The negative Fibonacci numbers alternate in sign and grow in magnitude as the index becomes more negative.
How to Calculate Negative Fibonacci Numbers
Calculating negative Fibonacci numbers involves a few key steps:
- Determine the absolute value of the negative index
- Calculate the standard Fibonacci number for that index
- Apply the sign based on the formula F-n = (-1)n+1 × Fn
For example, to calculate F-5:
- Absolute value: 5
- Standard Fibonacci F5 = 5
- Sign: (-1)5+1 = (-1)6 = 1
- Result: F-5 = 1 × 5 = 5
Our calculator automates this process for any negative integer input.
Applications of Negative Fibonacci Numbers
Negative Fibonacci numbers have several interesting applications in mathematics and computer science:
- Generating functions and combinatorics
- Number theory and Diophantine equations
- Algorithmic analysis and complexity
- Cryptography and sequence-based algorithms
The extended sequence helps in understanding the symmetry and periodicity of Fibonacci-like sequences, which is valuable in various mathematical proofs and computational problems.
Examples and Worked Solutions
Let's look at several examples of negative Fibonacci numbers:
| Index (n) | Fn | F-n | Calculation |
|---|---|---|---|
| 1 | 1 | -1 | F-1 = (-1)2 × 1 = -1 |
| 2 | 1 | 1 | F-2 = (-1)3 × 1 = -1 × 1 = -1 |
| 3 | 2 | 2 | F-3 = (-1)4 × 2 = 1 × 2 = 2 |
| 4 | 3 | -3 | F-4 = (-1)5 × 3 = -1 × 3 = -3 |
| 5 | 5 | 5 | F-5 = (-1)6 × 5 = 1 × 5 = 5 |
Notice the alternating pattern in the negative Fibonacci numbers. This pattern continues for all negative indices, maintaining the same relationship to the standard Fibonacci sequence.