Somos sequence

Example

For $k \ge 4$, the Somos-$k$ sequence is defined by $a_0 = a_1 = \cdots = a_{k-1} = 1$ and $$a_n\, a_{n-k} = \sum_{j=1}^{\lfloor k/2 \rfloor} a_{n-j}\, a_{n-k+j} \qquad (n \ge k).$$ This is a nonlinear difference equation: each new term is a quotient of a sum of products by $a_{n-k}$. For $k = 4$ the rule is $a_n = (a_{n-1} a_{n-3} + a_{n-2}^2)/a_{n-4}$. The Somos-4 sequence begins $1, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, \dots$. Check: $a_4 = (1 + 1)/1 = 2$, $a_5 = (2 \cdot 1 + 1)/1 = 3$, $a_6 = (3 \cdot 1 + 4)/1 = 7$, $a_7 = (7 \cdot 2 + 9)/1 = 23$, $a_8 = (23 \cdot 3 + 49)/2 = 59$, $a_9 = (59 \cdot 7 + 529)/3 = 314$. Although each step divides, the terms of Somos-$4$, $5$, $6$ and $7$ are all integers. For Somos-$8$ this fails: a non-integer term appears. Fomin and Zelevinsky showed more for $4 \le k \le 7$: with indeterminate initial values, each term is a Laurent polynomial in them with integer coefficients. For Somos-4 and Somos-5, this follows from the Laurent phenomenon for cluster algebras.