This question is motivated by integrability of the [sequence](https://mathoverflow.net/questions/143609/does-this-sequence-always-give-an-integer/202243#202243) mistakenly arisen in the question [Does this sequence always give an integer?](https://mathoverflow.net/questions/143609/does-this-sequence-always-give-an-integer)


Let $m\_1,\ldots, m\_{k-1}$ be positive integers and sequence $\{a\_n\}$ is defined by initial conditions
$$a\_1=\ldots=a\_k=1$$ and recurrence
$$a\_{n+k}a\_n=a\_{n+1}^{m\_1}a\_{n+k-1}^{m\_{k-1}}+a\_{n+2}^{m\_{2}}a\_{n+3}^{m\_{3}}\ldots a\_{n+k-2}^{m\_{k-2}}\quad(n\ge1).$$
Then the same (George Bergman's) argument described in David Gale's The strange and surprising saga of the Somos sequences, see [Tracking the automatic ant and other mathematical explorations](http://books.google.ru/books?id=wgVPu0YEOlQC&pg=PA1&hl=ru&source=gbs_toc_r&cad=3#v=onepage&q&f=false) allows to prove that $\{a\_n\}$ is an integer sequence.


**Question**: does this sequence belong to any known class of integer sequences?



On one class of Somos-like sequences
I believe this is a special case of Case (9) in Theorem 3.9 of [Allman, Cuenca and Huang](http://arxiv.org/abs/1309.0751). By the way, this paper was an REU project! 


