All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
If the compensation is to make up for an obligatory fast (like Ramadan), then it is obligatory. The expiatory fast is also obligatory. Fasting on alternate days like that of Prophet Daawood (David) is recommended (not obligatory). It is not valid to intend to observe an obligatory fast along with the intention of another obligatory fast – e.g. compensation and oath expiation – or to intend to observe an obligatory fast along with the intention of a voluntary fast –expiatory/compensatory and such optional fast as to fast every other day. One should, rather, fast a day for each: i.e. to fast for the compensation, then fast for the expiation, and then fast, if he so wishes, like the fasting of Daawood .
If you do not know the number of days that are due on you, then fast a number of days that you believe is most likely to be sufficient to free you from your liability.
Allah knows best.