69th Belarusian Mathematical Olympiad

Define the sequence $a_0,a_1,a_2,\dots$ 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.