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2020 EGMO P1: Linear recurrence is divisible by 2^2020

Source: 2020 EGMO P1

April 18, 2020
number theorySequenceEGMO

Problem Statement

The positive integers a0,a1,a2,…,a3030a_0, a_1, a_2, \ldots, a_{3030} satisfy 2an+2=an+1+4an for n=0,1,2,…,3028.2a_{n + 2} = a_{n + 1} + 4a_n \text{ for } n = 0, 1, 2, \ldots, 3028.
Prove that at least one of the numbers a0,a1,a2,…,a3030a_0, a_1, a_2, \ldots, a_{3030} is divisible by 220202^{2020}.
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