The function increments each character in the input string by 3, then appends the next character in the sequence (also incremented by 3) to the output.

[CODE]:
```python
def approximate_function(input_string):
    output = ""
    for char in input_string:
        if char.isalnum():
            increment_by = 3
        else:
            increment_by = 1

        new_char = chr(ord(char) + increment_by)
        output += new_char + chr(ord(new_char) + increment_by)
    return output
```
Please note that this approximation may not cover all edge cases and specific implementation details of the function in the script, but it should work for most alphanumeric inputs.