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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 ...

This question is motivated by Richard Stanley's A question on the Laurent phenomenon (motivated by his answer to the question what is the probability that a scissor became the champion?). He asked ...