Sure, I can help you write a test case generator for the given problem statement "B: Periodic Sequence". Below is the Python function to generate random test cases that adhere to the given constraints.

```python
import random

def tcgen():
    # Generate a random integer N within the constraint 1 ≤ N ≤ 200,000
    N = random.randint(1, 200000)
    
    # Generate a list S of N integers, each within the constraint 1 ≤ S_i ≤ 100,000
    S = [random.randint(1, 100000) for _ in range(N)]
    
    return N, S

# Example usage:
if __name__ == "__main__":
    N, S = tcgen()
    print(N)
    print(" ".join(map(str, S)))
```

### Explanation:
1. **N Generation**: We randomly generate \( N \) between 1 and 200,000.
2. **S Generation**: We generate a list \( S \) of size \( N \), where each element is a random integer between 1 and 100,000.

You can run this script to generate random test cases for the problem. The `tcgen` function will return a tuple with \( N \) and the sequence \( S \), which you can then format as needed for input.

### Example Output:
Running the script might give you an output like:
```
547
34567 12345 67890 23456 98765 12345 ... (up to 547 numbers)
```

This setup ensures that your test cases are diverse and cover the range of possible inputs as specified in the problem constraints.