(In reply to solution
Your result is correct, and it can be easily explained:
the number a(n) must cover all the qualifying subsets of n-1 integers i.e, a(n-1) plus all the qualifying subsets of n-2 elements, where to each of them integer n was added (the integer n-1 is absent!).
a(n)= a(n-1) + a(n-2)
a(1)=2=f(3) so a(n)=f(n+2)