# Problem Description
This is a geometric sequence problem involving nested squares and rotations. Starting with a unit square (n=1), each subsequent generation creates a pattern where the previous square is rotated 45 degrees and scaled in a specific way. The task is to calculate the area of the green region for any given generation n.

# Visual Facts
1. The sequence starts with a 1×1 white square (L=1)
2. The image shows 5 generations of the pattern
3. Each generation shows squares within squares
4. First square (n=1) is just a white square of side length 1
5. The second generation (n=2) shows a rotated square inside a larger square. 
6. The green area is the difference between the most inner square and its outer square, which consists of four triangles.
7. Every generation after n=1 has the previous square rotated by 45 degrees
8. The white square gets progressively smaller in each generation

# Visual Patterns
1. Mathematical Side Length Pattern of the Most Inner Square:
   - For n=1: Side length = 1
   - For n=2: Side length = 1/√2
   - For n=3: Side length = 1/2
   - For n=4: Side length = 1/(2√2)
   - For n=5: Side length = 1/4

2. General Patterns:
   - Each new square is formed by rotating the previous pattern by 45 degrees
   - The side length of each new inner square is related to the previous one by: S_n = S_(n-1)/√2
   - The ratio between consecutive square side lengths is constant: 1/√2
   - The area reduction follows a geometric sequence with ratio 1/2
   - General formula for side length appears to be: (1/2)^((n-1)/2)
   - When n = 1，the area of the green region is 0
   - When n>1, the area of the green region is equal to the area of the innermost rectangle: (1/2)^(n-1)

# Solution
def solution(n: int) -> float:
    """
    Calculates the area of the green region for a given generation level based on a specific pattern.

    Parameters:
        n (int): The generation level number.

    Returns:
        float: The area of the green region at generation level n.
    """
    if n == 1:
        return 0
    L = 1
    area = (L**2 / (2 ** (n - 2))) - (L**2 / (2 ** (n - 1)))
    
    return area


# Test Script
def test():
    test_cases = [
        (1, 0),
        (2, 0.5),
        (3, 0.25),
        (4, 0.125),
        (5, 0.0625),
        (6, 0.03125),
        (7, 0.015625),
        (8, 0.0078125),
        (9, 0.00390625),
        (10, 0.001953125)
    ]

    for n, expected_output in test_cases:
        try:
            output = solution(n)
        except Exception as e:
            error_message = (
                f"An exception occurred while running the test case:\n"
                f"`n`: {n}\n\n"
                f"Exception: '{str(e)}'"
            )
            assert False, error_message
            
        if output != expected_output:
            error_message = (
                f"The following Test Case failed:\n"
                f"`n`: {n}\n\n"
                f"Actual Output: {output}\n"
                f"Expected Output: {expected_output}"
            )
            assert False, error_message

test()