Well, Claire, this one seems screwy to me...but let me give you a start.
Make a table of differences...all the way out to the 6th difference which wil be -34.
You will see some patterns evolve. The only problem is in the left upper corner which I can't explain.
From the 2nd -1 in the initial sequence on, the diagonals in the difference table (running from SW to NE) are Fibonacci sequences, but the alternate diagonals have opposite signs and from a certain point on the columns in the difference table are Fibonacci sequences with alternating signs. These observations will allow you to predict the next 3 terms in the sequence as 3, -5, and 8.
I can't explain how this sequence was originally constructed, i.e. I cannot give you the general term. I will think about it some more and come back if I get anything else. Also when you get the answer from your instructor, I would appreciate your sending me a message to explain this problem better. Thanks. And good luck!