README.md

Fibonacci Sequence Visualization

This project contains a Python script to compute and visualize the Fibonacci sequence using recursion, memoization, and the matplotlib library for plotting. The code uses an efficient caching mechanism to store previously calculated Fibonacci numbers, speeding up the computation of higher values in the sequence.

Features

• Fibonacci Sequence Calculation: Computes the Fibonacci sequence up to the specified n.
• Memoization: Uses the lru_cache decorator from Python's functools module to cache computed Fibonacci numbers, improving performance.
• Visualization: Plots the sequence using matplotlib to illustrate the growth of Fibonacci numbers with increasing n.

Prerequisites

• Python 3.x
• matplotlib library: Install with pip install matplotlib if not already installed.

How to Use

1. Clone this repository or copy the script to your local machine.

2. Ensure matplotlib is installed in your Python environment.

3. Run the script with:

   python main.py

4. The program will display a plot showing the Fibonacci numbers for the first 10 values of n.

Code Explanation

• Caching with @lru_cache: This decorator stores results of previous function calls, reducing the need for repeated calculations. Here, @lru_cache(maxsize=1000) caches up to 1000 results of the fibonacci function.
• Recursive Function: The fibonacci(n) function uses recursion to calculate the nth Fibonacci number:
  • If n is 1 or 2, the result is 1 (base cases).
  • Otherwise, the result is the sum of the previous two Fibonacci numbers.
• Input Validation: Ensures n is a positive integer to prevent calculation errors.
• Plotting with matplotlib: Generates a line plot showing Fibonacci values against the sequence index n.

Example Output

The script plots the Fibonacci sequence values from n = 1 to n = 10, displaying the familiar exponential growth of the sequence.