2017 AMC 10A Problem 13

Let us list out the first few values to see if we can find a pattern in this sequence. $$0, 1, 1, 2, 0, 2, 2, 1, 0, 1, \cdots$$ The pair $(F_8,F_9)=(0,1)=(F_0,F_1).$ Because each later term is determined by the preceding two terms, the sequence therefore repeats every $8$ terms. The desired answer is the sum of $8$ consecutive numbers, which is fixed. This sum is $$0 + 1 + 1 + 2 + 0 + 2$$$$+ 2 + 1 = 9.$$ Thus, D is the correct answer.