TREY SONGZ and Ne-Yo are about to get a whole lot closer, because Trey’s dropping $35,000 a month on a sick new mansion in the Hollywood Hills -- right next door to his music-making collaborator.Trey will be renting the massive 13,000-square-foot crib starting April 1. We’ hear Ne-Yo’s pad isn’t quite as large. House envy?His 10-bedroom, 12-bath mega-house is a place you wouldn’t want to leave  ... floor-to-ceiling windows, a movie theater, gym, elevator, a slew of balconies, epic views and a rooftop pool ... where we assume all the magic will happen.And if the place looks familiar ... it’s because it’s one of four mansions -- all built by the same guy and all on the same street -- the four have been rented out for various shows, including "American Idol” and Playboy TV’s "Swing.”