A sequence of numbers is said to be weakly complete if every positive
integer
beyond a certain point is the sum of some subsequence
of
(Honsberger 1985). Dropping two terms from the Fibonacci
numbers produces a sequence which is not even weakly
complete. However, the sequence

is weakly complete, even with any finite subsequence deleted (Graham 1964).

Graham, R. "A Property of Fibonacci Numbers." Fib. Quart.2, 1-10, 1964.Honsberger, R. Mathematical
Gems III. Washington, DC: Math. Assoc. Amer., p. 128, 1985.