Do you have two keys programmed for the vehicle? I ask this because the car will program a new key itself, if you have two keys already programmed for the car.

 I would try to replace the battery in the key one more time, wait, I see you mentioned no lock/unlock so I would say that you have not told the programmed the pats, and somehow it got corrupted.

 When you place the key in the cupholder it doesn't need a battery. 

 I would replace the battery again and try it, if no go, I would get every key for the vehicle make sure they all have good batteries, then erase all keys for the vehicle, then program all keys to the vehicle, then access the pats and marry all the keys to the car. If somehow this doesn't work , then I will say you somehow have a bad receiver, because that's the only way the last thing does not work, if the keys are the correct freq.