# Distinct sums of Fibonacci numbers

The Fibonacci sequence is defined by $$F_1 = 1, F_2 = 1$$ and $$F_{n+2} = F_{n+1} + F_{n}$$ for $$n \geq 1$$.

Consider the set $$S_{15} = \{F_2, F_3, \ldots, F_{16}\}$$ of 15 terms of the sequence. If three distinct elements are chosen from the set $$S_{15}$$, how many different possible sums could these numbers have?

Details and assumptions

You may use the fact that $${15 \choose 3 } = 455$$.

Note that $$F_1$$ is not included in the set, so you can only use the number 1 once.

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