Here $F_n$ denote the $n$th Fibonacci number.
Is it possible that $2018=F_n-F_{a_1}-F_{a_2}-\dots - F_{a_k}$, where for any $i\leq k$ the inequality $F_{a_i+2}\geq F_n-F_{a_1}-\dots-F_{a_{i-1}}\geq 2F_{a_i}$ holds true?
So I tried to make a program, but it didn’t work. I will send it soon, not now because I am not at home. I believe there exist a mathematical proof, but I could’t find one. Please help! Anyway, probably the answer is no. Thanks in advance!