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Raving recurrence relation

Source: Philippine Mathematical Olympiad 2020/3

January 19, 2020

PMOnumber theory

Problem Statement

Define the sequence

\{a_i\}

a_0=1

a_1=4

, and

a_{n+1}=5a_n-a_{n-1}

for all

n\geq 1

. Show that all terms of the sequence are of the form

c^2+3d^2

for some integers

c

and

d