Geodesic
Polynomial Somos sequences II
Dalʹnevostočnyj matematičeskij žurnal, Tome 22 (2022) no. 1, pp. 91-99.

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It was proved in [1] that for $k=4,5,6,7$ the elements of the Somos-$k$ sequence defined by the recurrence $$S_k(n+k)S_k(n)=\sum_{1\leqslant i\leqslant k/2}\alpha_i x_0\dots x_{k-1}S_k(n+k-i)S_k(n+i)$$ and initial values $S_k(j)=x_j$ ($j=0,\dots,k-1$) are polynomials in the variables $x_0,\dots,x_{k-1}$. The unit powers of the variables $x_j$ in the factors \linebreak $\alpha_i x_0\dots x_{k-1}$ can be reduced. In this paper, we find the smallest values of these powers, at which the polynomiality of the above sequence is preserved. 
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M. A. Romanov. Polynomial Somos sequences II. Dalʹnevostočnyj matematičeskij žurnal, Tome 22 (2022) no. 1, pp. 91-99. http://geodesic.mathdoc.fr/item/DVMG_2022_22_1_a8/

[1] V. A. Bykovskii, M. A. Romanov, “Polinomialnye posledovatelnosti Somosa”, Funkts. analiz i ego pril., 55:1 (2021), 20–32 | MR | Zbl

[2] S. Fomin and A. Zelevinsky, “The Laurent Phenomenon”, Adv. Appl. Math., 28 (2002), 119–144 | DOI | MR | Zbl

[3] R. Robinson, “Periodicity of Somos sequences”, Proceedings of the AMS, 116:3 (1992), 613–619 | DOI | MR | Zbl

[4] Allan P. Fordy and Andrew Hone, “Symplectic Maps from Cluster Algebras”, Symmetry, Integrability and Geometry: Methods and Applications, 7 (2011), 091, 12 pp. | MR | Zbl

[5] Yoichi Nakata, The solution to the initial value problem for the ultradiscrete Somos-4 and 5 equations, 2017, 13 pp., arXiv: 1701.04262 | MR