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  "body": "\n## Example\n\nFor $k \\ge 4$, the *Somos-$k$ sequence* is defined by $a_0 = a_1 = \\cdots = a_{k-1} = 1$ and\n$$a_n\\, a_{n-k} = \\sum_{j=1}^{\\lfloor k/2 \\rfloor} a_{n-j}\\, a_{n-k+j} \\qquad (n \\ge k).$$\nThis is a [[nonlinear difference equation|7C7FBZGERM]]: each new term is a quotient of a sum of products by $a_{n-k}$.\nFor $k = 4$ the rule is $a_n = (a_{n-1} a_{n-3} + a_{n-2}^2)/a_{n-4}$.\nThe Somos-4 sequence begins $1, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, \\dots$.\nCheck: $a_4 = (1 + 1)/1 = 2$, $a_5 = (2 \\cdot 1 + 1)/1 = 3$, $a_6 = (3 \\cdot 1 + 4)/1 = 7$, $a_7 = (7 \\cdot 2 + 9)/1 = 23$, $a_8 = (23 \\cdot 3 + 49)/2 = 59$, $a_9 = (59 \\cdot 7 + 529)/3 = 314$.\nAlthough each step divides, the terms of Somos-$4$, $5$, $6$ and $7$ are all integers.\nFor Somos-$8$ this fails: a non-integer term appears.\nFomin 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.\nFor Somos-4 and Somos-5, this follows from the [[Laurent phenomenon|6YYPNKK6WZ]] for cluster algebras.\n\n## Explanations\n\n### L1 human\n\nSomos sequences are built by a rule that divides at every step, so one expects fractions to appear.\nSurprisingly, for the first few members of the family, every division comes out exact.\nThe reason is a hidden algebraic structure: the terms are Laurent polynomials in the starting values, so setting the starting values to $1$ gives integers.\n",
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