In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Somos sequence is a sequence of numbers defined by a certain
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
, 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 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
is defined by the equation
:
when ''k'' is odd, or by the analogous equation
:
when ''k'' is even, together with the initial values
: ''a''
''i'' = 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
:
while for ''k'' = 5 the equation is
:
These equations can be rearranged into the form of a
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
, 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
:
while for ''k'' = 5 it gives the recurrence
:
While in the usual definition of the Somos sequences, the values of ''a''
''i'' 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, ... .
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, ... .
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 algebra Cluster algebras are a class of commutative rings introduced by . A cluster algebra of rank ''n'' is an integral domain ''A'', together with some subsets of size ''n'' called clusters whose union generates the algebra ''A'' and which satisfy variou ...
s.
[.]
For ''k'' ≥ 8 the analogously defined sequences eventually contain fractional values. For Somos-8 the first fractional value is the 19th term with value 420514/7.
For ''k'' < 7, changing the initial values (but using the same recurrence relation) also typically results in fractional values.
References
External links
Jim Propp's Somos Sequence Site*{{mathworld, title=Somos Sequence, urlname=SomosSequence
The Troublemaker Number ''
Numberphile
''Numberphile'' is an educational YouTube channel featuring videos that explore topics from a variety of fields of mathematics. In the early days of the channel, each video focused on a specific number, but the channel has since expanded its s ...
'' video on the Somos sequences
Integer sequences
Recurrence relations