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x_{n + 2} = 3x_{n + 1}-2x_n, y_n = x^2_n+2^{n + 2} 2017 Ecuador NMO (OMEC) 3.5

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September 18, 2021
number theoryPerfect SquaresSequenceRecurrence

Problem Statement

Let the sequences (xn)(x_n) and (yn)(y_n) be defined by x0=0x_0 = 0, x1=1x_1 = 1, xn+2=3xn+1−2xnx_{n + 2} = 3x_{n + 1}-2x_n for n=0,1,...n = 0, 1, ... and yn=xn2+2n+2y_n = x^2_n+2^{n + 2} for n=0,1,...,n = 0, 1, ..., respectively. Show that for all n> 0, and n is the square of a odd integer.
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