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Problem 1 sequence of integer is square

Source: PAMO 2017 Problem 1

July 5, 2017

algebraPAMO

Problem Statement

We consider the real sequence

(x_n)

defined by

x_0=0, x_1=1

and

x_{n+2}=3x_{n+1}-2x_n

for

n=0,1,...

We define the sequence

(y_n)

by

y_n=x_n^2+2^{n+2}

for every non negative integer

n

. Prove that for every

n>0

,

y_n

is the square of an odd integer

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