2018 IMO Shortlist N3

Define the sequence $a_{0}, a_{1}, a_{2}, \ldots$ by $a_{n}=2^{n}+2^{\lfloor n / 2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way. (Serbia)

通过 $a_{n}=2^{n}+2^{\lfloor n / 2\rfloor}$ 定义序列 $a_{0}, a_{1}, a_{2}, \ldots$。证明存在无穷多个序列项可以表示为序列的(两个或多个)不同项之和，以及无穷多个不能以这种方式表达的项。 (塞尔维亚)