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Not eventually odd and not eventually even

Source: 2021 Taiwan TST Round 1 Independent Study 1-N

July 22, 2021
floor functionSequencenumber theoryTaiwan

Problem Statement

For each positive integer nn, define Vn=⌊2n2020⌋+⌊2n2021⌋V_n=\lfloor 2^n\sqrt{2020}\rfloor+\lfloor 2^n\sqrt{2021}\rfloor. Prove that, in the sequence V1,V2,…,V_1,V_2,\ldots, there are infinitely many odd integers, as well as infinitely many even integers.
Remark. ⌊x⌋\lfloor x\rfloor is the largest integer that does not exceed the real number xx.
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