How do I become friends with these people?

I’ve moved to a new city a few months ago and met a few really cool people through a roommate. I don’t really know many other people and I think that they all seem really nice; sometimes they invite me to hang out with them and most other times they do not, which is understandable since I am not really “in” with their group. But it is still frustrating when they do not invite me because after every time we do happen to hang out I thought I had left a good impression – made good conversation, got a few jokes in – all that good stuff. I also don’t want to be that guy who invites himself along.

I know that in most cases the answer to this would be for it to be my turn to invite them to do something fun, but being new, I don’t really know many fun things to do, and I honestly think it would come off as desperate – “hey group of friends that doesn’t really consider me their official friend, would you all care to join my lonesome self for a film/meal/etc?”

I am just very intimidated by the idea of trying to “break in” to a social circle that has already been established. Is there any way to go about doing this?