Disappearing Fibonacci

The traditional Fibonacci sequence starts with 0 and 1, and the rest of the terms are produced by a rule, which says that every term is equal to the sum of the previous two terms. You can produce other sequences by starting with different numbers but keeping the same rule. For example, you could start with 1 and 3, and produce this sequence:

1, 3, 4, 7, 11, 18, …

Can you come up with two starting numbers that make the resulting sequence converge to zero?

Caveats/Additional info: