Missing only one pill would not affect your protection if the contraception is established. Also, you do not normally need to use additional contraception if you're taking antibiotics, unless the antibiotics (or the illness being treated) cause diarrhoea or vomiting. Then absorption of the contraceptive pill may be affected. The exception would be if you are taking rifampicin-type antibiotics for a while.
For there to be any chance of becoming pregnant you would have had to have ovulated at least a week later than expected, but as sperm can live in the vagina for up to six days, ovulating twelve days later than expected could lead to pregnancy. So I would leave three weeks after last sex (1st Jan + 21 = 22nd Jan) for a test to be conclusive. However, I think there is very little chance you could be pregnant (even taking the BC pill under normal circumstances there is a very slight chance).
Why do you not have any BC at this time? If you have stopped the regime (you didn't say what BC pills), you may have to 'start again' (discuss this with your medical practitioner), and you may not be adequately protected now. Have you had sex at all apart from the dates you gave? If you have stopped taking the pills and you have none now, then follow the instructions on the leaflet when you get you new rx, but also discuss that with your medical practitioner. There are different types of BC pills.