Proof Writing & Problem Solving · Superchallenge problem

Using the Fibonacci numbers $1, 2, 3, 5, 8, 13, 21, 34, 55, 89$, each at most once, in how many ways can $100$ be written as a sum of DIFFERENT numbers from this list? (Order does not matter.) If two neighbouring numbers of the list are never used together, the representation is unique; without that rule it is not.