Given a multiset S of integers, when is $\sum_{s\in S}F_{n+s}=kF_{n+t}$ for some integers k and t and all integers n? $F_n$ is the n-th Fibonacci number.
Essentially, given a sum of Fibonacci numbers with fixed offsets, when can it be simplified to a multiple of a single Fibonacci number? (Example: $F_{n-1}+2F_{n}+F_{n+2}=2F_{n+2}.$)
The generalization $\sum_{s\in S}F_{n+s}=k_1F_{n+t_1}+k_2F_{n+t_2}$ is also of interest to me, if there is a nice N&S condition for it. And of course generalizations in other directions (any linear recurrence?) might be useful (though the special case of Fibonacci numbers is of special interest).