All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
There are signs that indicate the beginning of puberty. The first sign is the emission of sexual discharge by a man or a woman either in wakefulness or sleep. The second sign is the appearance of the coarse hair that grows on the pubic area. If any of these two signs appears, then one must have attained puberty even though one has not reached the age of 15 or 18 yet. The third sign that determines puberty is reaching the age of 15 according to Shafi‘i and Hanbali scholars and 18 according to Maliki scholars. According to Hanafi scholars, the age of puberty is 18 for males and 17 for females.
With this in mind, you are required to make up for the days that you have missed after having reached puberty. In addition, you are required to feed a poor person per day that you have missed because you delayed the immediate observance of these days.
If you cannot remember exactly how many days you have missed, then you should fast the amount of days of which you are almost sure that you have missed them, and that is enough for clearing you of your liability. You have mentioned in your question that the one who is unable to fast is required to feed sixty poor people, but this is not correct. However, if you remember that there was a day or days in which you broke your fast on account of engaging in sexual intercourse, then you are required to observe the expiation for breaking the fast by sexual intercourse.
Allaah Knows best.