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Two sequences imply a third - JBMO Shortlist

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October 30, 2010
algebra proposedalgebra

Problem Statement

Consider a sequence of positive integers xnx_n such that: (A) x2n+1=4xn+2n+2(\text{A})\ x_{2n+1}=4x_n+2n+2 (\text{B})\ x_{3n+\color[rgb]{0.9529,0.0980,0.0118}2}=3x_{n+1}+6x_n for all n≥0n\ge 0. Prove that (C) x3n−1=xn+2−2xn+1+10xn(\text{C})\ x_{3n-1}=x_{n+2}-2x_{n+1}+10x_n for all n≥0n\ge 0.
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