I'll let you think about it for a while longer before I say anything about your guess, but I think that recurrence would be X(N) = 7/18 x 3^N + 1/2. (blech!)
and while we're defining functions, how about we number coins 1,2,3,... and use
1,2,3 / 4,5,6 to mean weigh 1,2, and 3 against 4,5...