Find a recurrence relation for the number of ternary strings of length $n$ that contain a pair of consecutive $0$s
The answer to this can be found quite easily to be: $a_n=2a_{n-1}+2a_{n-2}+3^{n-2}\;.$
Now I came across a similar question which asked this:
Find a recurrence relation for the number of ternary strings of length $n$ that do NOT contain a pair of consecutive $0$s
I supposed that the answer would be the above answer subtracted from the total number os strings i.e $3^n$. But here on Pg 13 they're ending up with the same answer! They do start with an intuitive statement but they disregard it later.
What am I missing here?