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Sequence defined by quadratic recursion is integral

Source: KöMaL A. 738

February 13, 2019

number theoryalgebrarecurrence relation

Problem Statement

Consider the following sequence:

a_1 = 1

a_2 = 2

a_3 = 3

, and

a_{n+3} = \frac{a_{n+1}^2 + a_{n+2}^2 - 2}{a_n}

for all integers

n \ge 1

. Prove that every term of the sequence is a positive integer.