To generate test cases for the given problem, we need to consider the constraints and ensure that the generated test cases cover the edge cases, typical cases, and large input cases effectively. Here's how you can write a test case generator for the problem statement:

```python
import random

def tcgen():
    # Generate a random integer n within the given constraints
    n = random.randint(1, 10**6)
    
    return n

# Example usage:
if __name__ == "__main__":
    for _ in range(10):  # Generate 10 test cases
        print(tcgen())
```

This function generates a random integer \( n \) between 1 and 1,000,000, which is the constraint given in the problem statement. You can generate multiple test cases by invoking this function multiple times.

Here are some points to ensure while generating test cases:
1. Ensure that your test cases include both small values (e.g., 1, 2, etc.) and large values (close to 1,000,000).
2. Ensure some edge cases like the minimum (1) and the maximum (1,000,000) are included.

Given the nature of the problem where the output is based on the sequences that follow specific rules, the generated test cases are simply different values of \( n \) within the constraints.

You might also want to include some specific test cases manually, if needed, for thorough testing. For instance:
- Minimum edge case: `1`
- Small value: `2`
- Large value: `999999`
- Maximum edge case: `1000000`

This will help ensure that you are covering a wide range of possible inputs when testing your solution.