Get ready Part 28

Let $a_{m,n}$ denote the coefficient of $x^n$ in the expansion of $(1+x+x^2)^m$ . Is it true for all integers $k \ge 0$ , $0 \le \displaystyle \sum_{i=0}^{\lfloor \frac{2k}{3} \rfloor} (-1)^i a_{k-i,i} \le 1$ ?