Somos sequence

Recurrence equations

For an integer number k larger than 1, the Somos-k sequence (a_0, a_1, a_2, \ldots ) is defined by the equation :a_n a_{n-k} = a_{n-1} a_{n-k+1} + a_{n-2} a_{n-k+2} + \cdots + a_{n-(k-1)/2} a_{n-(k+1)/2} when k is odd, or by the analogous equation :a_n a_{n-k} = a_{n-1} a_{n-k+1} + a_{n-2} a_{n-k+2} + \cdots + (a_{n-k/2})^2 when k is even, together with the initial values : a**i = 1 for i

For k = 2 or 3, these recursions are very simple (there is no addition on the right-hand side) and they define the all-ones sequence (1, 1, 1, 1, 1, 1, ...). In the first nontrivial case, k = 4, the defining equation is :a_n a_{n-4} = a_{n-1} a_{n-3} + a_{n-2}^2 while for k = 5 the equation is :a_n a_{n-5} = a_{n-1} a_{n-4} + a_{n-2} a_{n-3},.

These equations can be rearranged into the form of a recurrence relation, in which the value a**n on the left hand side of the recurrence is defined by a formula on the right hand side, by dividing the formula by a**n − k. For k = 4, this yields the recurrence :a_n = \frac{a_{n-1} a_{n-3} + a_{n-2}^2}{a_{n-4}} while for k = 5 it gives the recurrence :a_n = \frac{a_{n-1} a_{n-4} + a_{n-2} a_{n-3}}{a_{n-5}}.

While in the usual definition of the Somos sequences, the values of a**i for i

Sequence values

The values in the Somos-4 sequence are :1, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, 8209, 83313, 620297, 7869898, ... . The values in the Somos-5 sequence are :1, 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, 6161, 22833, 165713, ... . The values in the Somos-6 sequence are :1, 1, 1, 1, 1, 1, 3, 5, 9, 23, 75, 421, 1103, 5047, 41783, 281527, ... . The values in the Somos-7 sequence are :1, 1, 1, 1, 1, 1, 1, 3, 5, 9, 17, 41, 137, 769, 1925, 7203, 34081, ... . The first 17 values in the Somos-8 sequence are :1, 1, 1, 1, 1, 1, 1, 1, 4, 7, 13, 25, 61, 187, 775, 5827, 14815 [the next value is fractional].{{citation | contribution-url = https://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B41/pdf/B41_003.pdf

Integrality

The form of the recurrences describing the Somos sequences involves divisions, making it appear likely that the sequences defined by these recurrence will contain fractional values. Nevertheless, for k ≤ 7 the Somos sequences contain only integer values. Several mathematicians have studied the problem of proving and explaining this integer property of the Somos sequences; it is closely related to the combinatorics of cluster algebras.{{citation

For k ≥ 8 the analogously defined sequences eventually contain fractional values. For Somos-8 the first fractional value is the 18th term with value 420514/7.

For k

References

References

1. Malouf, Janice L.. (1992). "An integer sequence from a rational recursion". [[Discrete Mathematics (journal).
2. "A Bare-Bones Chronology of Somos Sequences".
3. (2002). "The Laurent phenomenon". [[Advances in Applied Mathematics]].
4. (2004). "The Cube Recurrence". [[Electronic Journal of Combinatorics]].
5. Stone, Alex. (18 November 2023). "The Astonishing Behavior of Recursive Sequences".