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a_{n+4}=2a_{n+3}+a_{n+2}-2a_{n+1}-a_n, prove n^2 / a_n for infinite n

Source: Balkan BMO Shortlist 2015 N2

August 5, 2019
recurrence relationnumber theory with sequencesSequencedividesnumber theoryalgebra

Problem Statement

Sequence (an)n≥0(a_n)_{n\geq 0} is defined as a0=0,a1=1,a2=2,a3=6a_{0}=0, a_1=1, a_2=2, a_3=6, and an+4=2an+3+an+2−2an+1−an,n≥0 a_{n+4}=2a_{n+3}+a_{n+2}-2a_{n+1}-a_n, n\geq 0. Prove that n2n^2 divides ana_n for infinite nn.
(Romania)
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