The following is a problem we were unable to prove and left stated in the paper "Arithmetical properties of a sequence arising from an arctangent sum", J. Numb. Theory 128 (2008) 1807–1846.
Define the sequence $\{x_n\}$ by $x_1=1$ and for $n\geq2$ by $$x_n=\frac{x_{n-1}+n}{1-nx_{n-1}}.$$ We believe and claim that $x_n$ is not an integer, for $n\geq5$. Any ideas?
Note: $x_1=1, x_2=-3, x_3=0, x_4=4$. Among other things, we proved $x_n\neq0$ for $n\geq4$.
Special case: Since the problem is a bit hard and in view of no success being reported, I propose the following special case to attend to: prove (or disprove, which I doubt) that $$x_n\neq 1 \qquad \text{for any $n\geq2$}.$$