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a_{n+1} = a_n + 2 \cdot 3^n , find rational a_o such a^j_k / a^k_j

Source: 2014 Austrian Federal Competition For Advanced Students, Part 1 p3

February 2, 2020

recurrence relationSequencenumber theory

Problem Statement

Let

a_n

be a sequence defined by some

a_0

and the recursion

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

for

n \ge 0

. Determine all rational values of

a_0

such that

a^j_k / a^k_j

is an integer for all integers

j

and

k

with

0 < j < k