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Expressing a term of the sequence by the sum of previous terms

Source: ISL 2018 N3

July 17, 2019
IMO Shortlistnumber theory

Problem Statement

Define the sequence a_0,a_1,a_2,\hdots by an=2n+2⌊n/2⌋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.
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