For the second sum let $\omega=e^{2\pi i /3}$ be a third root of unity. The binomial expansion tells us
$$\left ( 1+x \right )^n = \sum_{k=0}^{n}\binom{n}{k}x^n$$
Subbing $x$ with $1, \omega, \omega^2$ respectively at the binomial expansion we get:
$$\begin{matrix}
\displaystyle \sum_{k=0}^{n}\binom{n}{k}=2^n & , &\displaystyle \left ( 1+\omega \right )^n =\sum_{k=0}^{n}\binom{n}{k}\omega^n &, &\displaystyle \left ( 1+\omega^2 \right )^n = \sum_{k=0}^{n} \binom{n}{k}\omega^{2k}
\end{matrix}$$
Adding the above three equations together we get that:
$$\sum_{k=0}^{n}\binom{n}{k}\left [ 1+\omega^k +\omega^{2k} \right ]=2^n + \left ( 1+\omega \right )^n + \left ( 1+\omega^2 \right )^n $$
We note that $1+\omega^k +\omega^{2k}=3$ when $k$ is divisible by $3$ and $0$ othewise. Hence:
$$\sum_{m=0}^{\left \lfloor n/3 \right \rfloor}\binom{n}{3m}= \frac{1}{3}\left [ 2^n + \left ( 1+\omega \right )^n+ \left ( 1+\omega^2 \right )^n \right ]$$
Subbing $n$ with $3n$ and doing the calculations we finally get that:
$$\sum_{k=0}^{n}\binom{3n}{3k}= \frac{1}{3}\left [ 8^n + 2(-1)^n \right ]$$
The first sum is much easier and I leave it to someone else.
