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Find all a which generates squares in the sequence

Source: Baltic Way 2005

December 28, 2010
number theory proposednumber theory

Problem Statement

A sequence (xn)n≥0(x_n)_{n\ge 0} is defined as follows: x0=a,x1=2x_0=a,x_1=2 and xn=2xn−1xn−2−xn−1−xn−2+1x_n=2x_{n-1}x_{n-2}-x_{n-1}-x_{n-2}+1 for all n>1n>1. Find all integers aa such that 2x3n−12x_{3n}-1 is a perfect square for all n≥1n\ge 1.
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