Binary password

Jomart uses a binary string as the password for his computer. Recently, he forgot his old password and now wants to obtain a new one, which will be a binary string of length $n$. He considers a password sufficiently secure if it does not contain two consecutive zeros.

To obtain a new password, Jomart generates a random binary string of length $n$. If the string is not secure, he generates another one and repeats the process until he obtains a secure password.

Find the expected number of randomly generated passwords Jomart will need before he finds a secure one.