Seemingly simple decryption question
If I interpret your question correctly, your unchanging 4-character input is probably just a PIN code that unlocks the use of the device. In this case, the sequence of 6-character strings is a series of one-time password to unlock access to a remote device.

Under this assumption, they is no simple way to predict the rest of the sequence. Indeed, it is easy to design secure device of this type. A simple solution is to output a hash of some inner state and then update the state using a secret key algorithm keyed with a key shared by the device and its remote partner. The most difficult issue is to solve desynchronization problems (e.g. what happens if you advance your local device 1000 times without transmitting anything, will the 1001-th code work ?)

Of course, it is also possible to construct insecure devices of this type. So if you have luck your exact device might be attackable.

However, the short answer is: no, in general, this is not doable.