From the form of their defining recurrence (which involves division), one would expect the terms of the sequence to be fractions, but surprisingly, a few Somos sequences have the property that all of their members are integers.
is defined by the equation when k is odd, or by the analogous equation when k is even, together with the initial values 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 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 while for k = 5 it gives the recurrence 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.
[2][3][4] 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 algebras.
[5][3][6][7] For k ≥ 8 the analogously defined sequences eventually contain fractional values.