Fibonacci Calculator

How the Fibonacci Calculator Works:

With this tool:

• Single Fibonacci Number Calculation: You can compute the Fibonacci number for a given n.
• Fibonacci Sequence Calculation: You can generate a sequence of Fibonacci numbers between two values, n1 and n2.
• The calculator supports numbers in the range of -200 to 200 for n.

Fibonacci Formula:

For positive values of n, the Fibonacci sequence is defined recursively by the following relation:

$$F_{n} = F_{n-1} + F_{n-2}$$

where the initial values are:

$$F_0 = 0 \quad \text{and} \quad F_1 = 1$$

For negative values of n, the Fibonacci sequence is defined as:

$$F_{-n} = (-1)^{n+1} F_{n}$$

This ensures that the Fibonacci sequence maintains its recursive structure but alternates in sign for negative values of n.

Examples

1. Single Fibonacci Number Calculation:

   • Input: $n = 10$
     
   • Formula: $$F_{10} = F_9 + F_8$$
   • Calculation: $$F_9 = 34, \quad F_8 = 21 \quad \text{so} \quad F_{10} = 34 + 21 = 55$$
   • Result: $F_{10} = 55$
     

2. Fibonacci Sequence Calculation:

   • Input: $n_1 = 4$, $n_2 = 7$
     
   • Formula for each Fibonacci number: $$F_4 = F_3 + F_2 = 2 + 1 = 3$$$$F_5 = F_4 + F_3 = 3 + 2 = 5$$$$F_6=F_5+F_4=5+3=8$$$$F_7=F_6+F_5=8+5=13$$
   • Result: $$F_4=3,F_5=5,F_6=8,F_7=13$$
   • Sequence: $3, 5, 8, 13$
     

3. Negative Fibonacci Number Calculation:

   • Input: $n = -5$
     
   • Formula: $$F_{-5}=(-1{)}^{5+1}{F}_5=-1\times 5=-5$$
   • Result: $F_{-5} = -5$
     

Key Features:

1. Custom Result Format: You can format the Fibonacci sequence as a table, list, or separated by commas, spaces, or new lines.
2. Thousands Separator: For large Fibonacci numbers, separators (such as commas or periods) can be applied to enhance readability.

Quick Tips:

• Range for n: Input values must be between -200 and 200.
• Sign of Negative Fibonacci Numbers: Negative Fibonacci numbers alternate in sign, following the formula for $F_{-n}$.