Are there any sequence given by a recurrence relation:
$x_{n+t}=P(x_t,\cdots,x_{t+n-1})$, where $P$ is a *positive Laurent Polynomial*, satisfy:

if $x_0=\cdots=x_{n-1}=1$, then the sequence is only integer;

but does not exhibit Laurent Phenomenon ?

What if we allow $P$ to be a rational function?