This question is motivated by integrability of the sequence mistakenly arisen in the question 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 allows to prove that $\{a_n\}$ is an integer sequence.

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