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PAMO Problem 1: Linear recurrence relation

Source: 2019 Pan-African Mathematics Olympiad, Problem 1

April 9, 2019
recurrence relationLinear RecurrencesalgebrainductionPAMO

Problem Statement

Let (an)n=0∞(a_n)_{n=0}^{\infty} be a sequence of real numbers defined as follows:
[*] a0=3a_0 = 3, a1=2a_1 = 2, and a2=12a_2 = 12; and [*] 2an+3−an+2−8an+1+4an=02a_{n + 3} - a_{n + 2} - 8a_{n + 1} + 4a_n = 0 for n≥0n \geq 0.
Show that ana_n is always a strictly positive integer.
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