Somos sequence
Encyclopedia
In mathematics
, a Somos sequence is a sequence of numbers defined by a bilinear recurrence relation, 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.
when k is odd, or by the analogous equation
when k is even.
For instance, for 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 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 sn − k. For k = 4, this yields the recurrence
while for k = 5 it gives the recurrence
In the usual definition of the Somos sequences, the values of ai for i < k are all set equal to one, as in the recurrence for the Fibonacci sequence. However it is also possible to define other sequences by using the same recurrences with different initial values.
The values in the Somos-5 sequence are
The values in the Somos-6 sequence are
The values in the Somos-7 sequence are
For k ≥ 8 the analogously defined sequences eventually contain fractional values.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Somos sequence is a sequence of numbers defined by a bilinear recurrence relation, discovered by mathematician Michael Somos
Michael Somos
Michael Somos is an American mathematician, currently a visiting scholar in Georgetown University. In the late eighties he proposed an intriguing conjecture about certain polynomial recurrences, now called Somos sequences, that surprisingly in some cases contain only integers...
. 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
A Somos-k sequence, for an integer number k, is defined by the equationwhen k is odd, or by the analogous equation
when k is even.
For instance, for 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
Recurrence relation
In mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms....
, 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 sn − k. For k = 4, this yields the recurrence
while for k = 5 it gives the recurrence
In the usual definition of the Somos sequences, the values of ai for i < k are all set equal to one, as in the recurrence for the Fibonacci sequence. However 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.For k ≥ 8 the analogously defined sequences eventually contain fractional values.