Aaron is trying to write a program to compute the terms of the sequence defined recursively by a_{0}=0, a_{1}=1, and
a_{n}= \begin{cases}a_{n-1}-a_{n-2} & n \equiv 0 \\ 2 a_{n-1}-a_{n-2} & \text { else }\end{cases}
However, Aaron makes a typo, accidentally computing the recurrence by
a_{n}= \begin{cases}a_{n-1}-a_{n-2} & n \equiv 0 \\ 2 a_{n-1}-a_{n-2} & \text { else }\end{cases}
For how many 0 \leq k \leq 2016 did Aaron coincidentally compute the correct value of a_{k}?