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 $(a_0, a_1, a_2, \ldots )$ 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

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

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

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 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

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

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 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 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 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 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 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), Advances in Applied Mathematics 28: 119–144, arXiv:math.CO/0104241 |chapter= ignored (help).
3. ^ Carroll, Gabriel D.; Speyer, David E. (2004), Electronic Journal of Combinatorics 11: R73, arXiv:math.CO/0403417 |chapter= ignored (help).