advanced counting: bit strings with 2x '00'

how many n-length bit strings are there, where '00' occurs exactly **two **times. ( '000' is perceived as **two** occurings )

my try:

we have to distinguish between different kinds of bit strings:

1. ..00..00..

2. ..000...

anything else is not a valid bitstring in our example.

it may has something to do with this: id:A055243 - OEIS Search Results

first formulas for solving this are posted here: MP-Forum: Anzahl der Binärwörter mit genau 2-mal "00" (Matroids Matheplanet) ( german )

this is a homework in our discrete mathematics class, i talked with about 20 people who tried to solve this for the last 4 days, no one did. and i am starting to get depressed myself : /