Ternary strings (combinatorics, recurrence) The questions is: for $A_n$ find all ternary strings of length $n ≥ 0$ that don't include substring $”11”$. Provide answer in form of:
a) recurrence relation
b) combinatorial expression 
After that, for $B_n$ take $A_n$ and exclude strings that also have substring $”12”$ and end with $”1”$ (at the same time).
The biggest issue for me comes with combinatorial expression, whatever I try I cannot include all variations and get kind of lost. Might appreciate a bit of help on recurrence relation as well. 

I happened to chance on this, so here is the combinatorial approach that you wanted, which is incidentally much simpler !
Hope you like it !
Combinatorial approach
Any string of $i$ non-$1's$ create $(i+1)$ gaps (including ends) where non-adjacent $1's$ can be put,
thus if string length is $n$, the minimum number of non $1's$ needed is $\lfloor\frac{n}2\rfloor$ to ensure that no $1's$ are adjacent.
and in the $(i+1)$ gaps, we just need to fit in the $(n-i)\;\;1's$
$$f(n) = \sum_{i={\lfloor\frac{n}2\rfloor}}^n\binom{i+1}{n-i}$$
