probability - Expected number of tosses before you see a repeat.

Suppose we roll a fair die until some face has appeared twice. For instance, we might have a run of rolls 12545 or 636. How many rolls on average would we make? What if we roll until a face has appeared three times?

I calculated the expected value for getting a repeat for a six-sided dice and I got 1223/324 (3.77 tosses). How would we generalize this to an $n$-sided dice? What about $k$-repeats?