MathDB

a_n = 14a_{n-1} + a_{n-2} , b_n = 6b_{n-1}-b_{n-2}, infinite common elements

Source: 2019 Austrian Federal Competition For Advanced Students, Part 1 p1

March 4, 2020

Sequencerecurrence relationRecurrencealgebraAustria

Problem Statement

We consider the two sequences

(a_n)_{n\ge 0}

and

(b_n) _{n\ge 0}

of integers, which are given by

a_0 = b_0 = 2

and

a_1= b_1 = 14

and for

n\ge 2

they are defined as

a_n = 14a_{n-1} + a_{n-2}

b_n = 6b_{n-1}-b_{n-2}

. Determine whether there are infinite numbers that occur in both sequences