South African Mathematics Olympiad First Round

The first six terms of the sequence are $3$, $3$, $6$, $9$, $15$, $24$, $...$, which are Odd, Odd, Even, Odd, Odd, Even, $...$, since Odd $+$ Odd $=$ Even and Odd $+$ Even $=$ Odd. The pattern continues in the same way in cycles of length $3$, with two odd numbers and one even number in each cycle. After division by $2$, the remainders in each cycle are $1$, $1$, $0$, so the sum of the three remainders is $2$. The sum of the first $2016$ remainders is therefore $\frac{2}{3}\times 2016=1344$.