International Mathematics Competition
for University Students
2019

IMC2019: Day 1, Problem 4

Problem 4. Define the sequence \(\displaystyle a_0,a_1,\ldots\) of numbers by the following recurrence:

\(\displaystyle a_0=1, \quad a_1=2, \quad (n+3)a_{n+2}=(6n+9)a_{n+1}- na_n \quad \text{for \(\displaystyle n\ge 0\).} \)

Prove that all terms of this sequence are integers.

Proposed by Khakimboy Egamberganov, ICTP, Italy

Hint: Determine the generating function \(\displaystyle \sum a_nx^n\).

  Solution