Certainly! Given the problem statement for the "Flick Input" problem, we need to create a test case generator that can produce random but valid inputs for this competitive programming problem. Here's how we can construct the test case generator:

```python
import random

def tcgen():
    # Define the length of the input string (2 to 1000 characters)
    length = random.randint(2, 1000)

    # Define possible digits and flick directions
    digits = '0123456789'
    flicks = 'TLURD'

    # Generate a random sequence of operations
    input_str = ''.join(random.choice(digits) + random.choice(flicks) for _ in range(length // 2))
    
    # If the length is odd, add one more digit-flick pair
    if length % 2 == 1:
        input_str += random.choice(digits) + random.choice(flicks)

    return input_str

# Example of generating a test case
print(tcgen())
```

### Explanation:
1. **Input Length**: The length of the input string is chosen randomly between 2 and 1000.
2. **Digits and Flick Directions**: The digits `0-9` and flick directions `T, L, U, R, D` are defined.
3. **Generating Input String**: 
   - A loop runs for half the length (since each operation is a digit followed by a flick direction).
   - For each iteration, a random digit and a random flick direction are concatenated.
   - If the length is odd, an additional digit-flick pair is added to ensure the total length is within the valid range.

This generator will produce random yet valid test cases for the given problem. You can call `tcgen()` to get a new test case each time you run it.

### Example Output:
```
3T7L0D8U
```
This output represents a sequence of flick operations starting with touching button 3, flicking button 7 to the left, and so on.

This test case generator ensures that the input is always within the constraints specified in the problem statement and provides a diverse set of inputs to thoroughly test the solution.