# Somos sequence

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. It is not obvious from the form of their defining recurrence that every number in a Somos sequence is an integer, but nevertheless many Somos sequences have the property that all of their members are integers.

## Recurrence equations

For an integer number k larger than 1, the Somos-k sequence ${\displaystyle (a_{0},a_{1},a_{2},\ldots )}$ is defined by the equation

${\displaystyle a_{n}a_{n-k}=a_{n-1}a_{n-k+1}+a_{n-2}a_{n-k+2}+\cdots +a_{n-(k-1)/2}a_{n-(k+1)/2}}$

when k is odd, or by the analogous equation

${\displaystyle a_{n}a_{n-k}=a_{n-1}a_{n-k+1}+a_{n-2}a_{n-k+2}+\cdots +(a_{n-k/2})^{2}}$

when k is even, together with the initial values

ai = 1 for i < k.

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

${\displaystyle a_{n}a_{n-4}=a_{n-1}a_{n-3}+a_{n-2}^{2}}$

while for k = 5 the equation is

${\displaystyle a_{n}a_{n-5}=a_{n-1}a_{n-4}+a_{n-2}a_{n-3}\,.}$

These equations can be rearranged into the form of a recurrence relation, in which the value an on the left hand side of the recurrence is defined by a formula on the right hand side, by dividing the formula by an − k. For k = 4, this yields the recurrence

${\displaystyle a_{n}={\frac {a_{n-1}a_{n-3}+a_{n-2}^{2}}{a_{n-4}}}}$

while for k = 5 it gives the recurrence

${\displaystyle a_{n}={\frac {a_{n-1}a_{n-4}+a_{n-2}a_{n-3}}{a_{n-5}}}.}$

While in the usual definition of the Somos sequences, the values of ai for i < k are all set equal to 1, it is also possible to define other sequences by using the same recurrences with different initial values.

## 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, ... (sequence A006720 in the OEIS).

The values in the Somos-5 sequence are

1, 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, 6161, 22833, 165713, ... (sequence A006721 in the OEIS).

The values in the Somos-6 sequence are

1, 1, 1, 1, 1, 1, 3, 5, 9, 23, 75, 421, 1103, 5047, 41783, 281527, ... (sequence A006722 in the OEIS).

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, ... (sequence A006723 in the OEIS).

## 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.[1][2][3] An elementary proof for integrality of the Somos-5 sequence by Michael J Crabb of Glasgow University may be found in Michael Wemyss's website [1].

For k ≥ 8 the analogously defined sequences eventually contain fractional values.

For k < 7, changing the initial values (but using the same recurrence relation) also typically results in fractional values.

## References

1. ^ Malouf, Janice L. (1992), "An integer sequence from a rational recursion", Discrete Mathematics, 110 (1–3): 257–261, doi:10.1016/0012-365X(92)90714-Q.
2. ^ Fomin, Sergey; Zelevinsky, Andrei (2002), "The Laurent phenomenon", Advances in Applied Mathematics, 28: 119–144, arXiv:.
3. ^ Carroll, Gabriel D.; Speyer, David E. (2004), "The Cube Recurrence", Electronic Journal of Combinatorics, 11: R73, arXiv:.