Find the closed expression for binomial sum

Question

I need help in finding closed expression for the following sum: $a_n = \sum\limits_{k=0}^n (-1)^k \binom{2n - k}{k}$. By inspecting the first several elements of the sequence I came up with the hypothesis for expression for $a_n$: $a_{3j} = 1,\space a_{3j+1} = 0,\space a_{3j+2} = -1,$ where $j=0,1,2,\dots.$ I was not able to prove this by induction. Is it an easy exercise? What other approaches can be taken to solve this problem? I am mostly looking for as basic solutions as possible since I am new to combinatorics.