Laurent phenomenon sequences

In this paper, we undertake a systematic study of sequences generated by recurrences \(x_{m+n}x_m = P(x_{m+1}, \ldots , x_{m+n-1})\) which exhibit the Laurent phenomenon. Some of the most famous among these are the Somos and the Gale-Robinson sequences. Our approach is based on finding period 1 seeds of Laurent phenomenon algebras of Lam–Pylyavskyy. We completely classify polynomials P that generate period 1 seeds in the cases of \(n=2,3\) and of mutual binomial seeds. We also find several other interesting families of polynomials P whose generated sequences exhibit the Laurent phenomenon. Our classification for binomial seeds is a direct generalization of a result by Fordy and Marsh, that employs a new combinatorial gadget we call a double quiver.