2020 Putnam A5

Let $a_n$ be the number of sets $S$ of positive integers for which

$$

\sum_{k \in S} F_k = n,

$$

where the Fibonacci sequence $(F_k)_{k \geq 1}$ satisfies $F_{k+2} = F_{k+1} + F_k$ and begins $F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$. Find the largest integer $n$ such that $a_n = 2020$.

设 $a_n$ 为正整数集合 $S$ 的数量，其中

$$

\sum_{k \in S} F_k = n，

$$

其中斐波那契数列 $(F_k)_{k \geq 1}$ 满足 $F_{k+2} = F_{k+1} + F_k$ 并开始 $F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$。找到满足 $a_n = 2020$ 的最大整数 $n$。