How many binary words of length n , that consist an even number of zeros? How many binary words (chars '0' and or '1') of length n that consist an even number of zeros are there?
I know that there are $2^n$ options overall, and that for every $n$, there are $\lceil{\frac{n}{2}}\rceil + 1$ options for even zeros. But now what? I got lost! 
Maybe I need to use the pascal triangle?

Hints:
Let $a_n$ be the number that you are looking for.
If $n$ is odd, there are as many words with an even number of zeros as many words with an odd number of zeros, because the complementary words of the former are exactly the latter. 
If $n$ is even, then you can split the words in half, and if the word has an even number of zeros then both halves have an odd number of zeros or both even, so $a_n=2a_{n/2}^2$.
