**Putnam 2017 A2**

Prove that the sequence of rational functions defined by

\begin{align}
Q_0 &=\ 1 \\
Q_1 &=\ x \\
Q_n &=\ \frac{Q_{n-1}^2-1}{Q_{n-2}}
\end{align}

is actually a sequence of polynomials with integer coefficients.

**Solution (using a determinant property)**

Define
\begin{align}
P_0 &=\ 1 \\
P_1 &=\ x \\
P_n &=\ xP_{n-1} - P_{n-2}
\end{align}

Clearly \(\{P_n\}\) is a sequence of polynomials with integer coefficients.
Now show that \(\{P_n\}\) and \(\{Q_n\}\) define the same sequence:

\begin{align}
&\ \ -P_nP_{n-2} +P_{n-1}^2
\\&=\
\det\begin{pmatrix}
P_n & -P_{n-1} \\ P_{n-1} & -P_{n-2}
\end{pmatrix}
\\&=\
\det\begin{pmatrix}
x & -1 \\
1 & 0
\end{pmatrix}^n
\\&=\ 1
\end{align}

This shows that
\begin{equation}
P_n = \frac{P_{n-1}^2-1}{P_{n-2}}
\end{equation}