Sure! Below is a test case generator for the given problem statement about finding Shichi-Go-San numbers.

```python
import random

def tcgen():
    # Generate a random integer N in the range [1, 10^9)
    N = random.randint(1, 10**9 - 1)
    
    return N

# Example of generating a test case
if __name__ == "__main__":
    print(tcgen())
```

### Explanation:
- The function `tcgen()` generates a single random integer `N` within the specified range `[1, 10^9)`.
- The code makes use of Python's `random.randint` function to ensure that `N` is within the constraints provided (1 ≤ `N` < 10^9).
- The generated test case will be a valid input for the problem statement.

Here's how `tcgen()` can be used to generate multiple test cases:

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

This will print 10 different integers, each suitable for testing the solution to the Shichi-Go-San number counting problem.