Somos sequence

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

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[edit]

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[edit]

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[edit]

  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).

External links[edit]