Hint:

Let's say we want to find . For the sake of brevity, let's say a string is "acceptable" if it does not contain two adjacent vowels.

Break the possibilities into two disjoint sets depending on whether the (n+1)-th letter is a vowel. The total number of possibilities is the sum of the sizes of these sets.

If the (n+1)-th letter is a vowel, the preceding n letters can be any acceptable string which does not end in a vowel. If the nth letter is not a vowel, then the preceding n-1 letters can be any acceptable string, which can be done in ways. Then how many choices are there for the nth and (n+1)-th letters?

If the (n+1)-letter is not a vowel, then the preceding n letters can be any acceptable string, which can be done in ways. Then how many choices are there for the (n+1)-th letter?