In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below. They were discovered by mathematician Michael Somos. From the form of their defining recurrence (which involves division), one would expect the terms of the sequence to be fractions, but surprisingly, a few Somos sequences have the property that all of their members are integers.
Recurrence equations For an integer number k {\displaystyle k} larger than 1 {\displaystyle 1} , a Somos- k {\displaystyle k} sequence { a n } n ∈ Z {\displaystyle \{a_{n}\}_{n\in \mathbb {Z} }} is a solution of the equation
a n a n − k = ∑ i = 1 ⌊ k / 2 ⌋ α i a n − i a n − k + i {\displaystyle a_{n}a_{n-k}=\sum _{i=1}^{\lfloor k/2\rfloor }\alpha _{i}a_{n-i}a_{n-k+i}}
where α 1 {\displaystyle \alpha _{1}} , α 2 {\displaystyle \alpha _{2}} , ..., α ⌊ k / 2 ⌋ {\displaystyle \alpha _{\lfloor k/2\rfloor }} are fixed parameters. It can be rearranged into the form of an order k {\displaystyle k} recurrence relation
a n = ∑ i = 1 ⌊ k / 2 ⌋ α i a n − i a n − k + i a n − k . {\displaystyle a_{n}={\frac {\sum _{i=1}^{\lfloor k/2\rfloor }\alpha _{i}a_{n-i}a_{n-k+i}}{a_{n-k}}}.} Hence a non-degenerate solution is determined by a choice of k {\displaystyle k} initial values a 0 {\displaystyle a_{0}} , a 1 {\displaystyle a_{1}} , ..., a k − 1 {\displaystyle a_{k-1}} . The sequence yielded by setting α 1 = α 2 = ⋯ = α ⌊ k / 2 ⌋ = 1 {\displaystyle \alpha _{1}=\alpha _{2}=\dots =\alpha _{\lfloor k/2\rfloor }=1} and a 0 = a 1 = ⋯ = a k − 1 = 1 {\displaystyle a_{0}=a_{1}=\dots =a_{k-1}=1} is referred to as the Somos- k {\displaystyle k} sequence. The Somos- k {\displaystyle k} sequence is symmetric, i.e., s − n = s n + k − 1 {\displaystyle s_{-n}=s_{n+k-1}} . For k = 2 {\displaystyle k=2} or 3 {\displaystyle 3} , the defining relations are very simple (there is no addition on the right-hand side). In the first nontrivial case, k = 4 {\displaystyle k=4} , the relation is
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