2004 Putnam A3

Define a sequence $\{ u_n \}_{n=0}^\infty$

by $u_0 = u_1 = u_2 = 1$, and thereafter by

the

condition that

$$

\det\begin{pmatrix}

u_n & u_{n+1}\\

u_{n+2} & u_{n+3}

\end{pmatrix}

= n!

$$

for all $n \ge 0$. Show that $u_n$ is an integer for all $n$.

(By convention, $0! = 1$.)

定义序列 $\{ u_n \}_{n=0}^\infty$

通过 $u_0 = u_1 = u_2 = 1$，然后通过

的

条件是

$$

\det\begin{pmatrix}

u_n & u_{n+1}\\

u_{n+2} 和 u_{n+3}

\end{p 矩阵}

= n！

$$

对于所有 $n \ge 0$。证明 $u_n$ 是所有 $n$ 的整数。

(按照惯例，$0！= 1$。)