The Somos sequences are a set of related symmetrical recurrence relations which, surprisingly, always give integers. The Somos sequence of order
, or Somos-
sequence, is defined by
 |
(1)
|
where 
is the floor function and 
for 
, ..., 
.
The 2- and 3-Somos sequences consist entirely of 1s. The 
-Somos sequences for 
, 5, 6, and 7 are
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  | ![1/(a_(n-6))[a_(n-1)a_(n-5)+a_(n-2)a_(n-4)+a_(n-3)^2]](/images/equations/SomosSequence/Inline17.svg) |
(4)
|
 |  | ![1/(a_(n-7))[a_(n-1)a_(n-6)+a_(n-2)a_(n-5)+a_(n-3)a_(n-4)].](/images/equations/SomosSequence/Inline20.svg) |
(5)
|
The first few terms are summarized in the following table.
 | OEIS |  ,  , ... |
| 4 | A006720 | 1, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, ... |
| 5 | A006721 | 1, 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, ... |
| 6 | A006722 | 1, 1, 1, 1, 1, 1, 3, 5, 9, 23, 75, 421, 1103, ... |
| 7 | A006723 | 1, 1, 1, 1, 1, 1, 1, 3, 5, 9, 17, 41, 137, 769, ... |
Combinatorial interpretations for Somos-4 and Somos-5 were found by Speyer (2004) and for Somos-6 and Somos-7 by Carroll and Speyer (2004).
Gale (1991) gives simple proofs of the integer-only property of the Somos-4 and Somos-5 sequences, and attributes the first proof to Janice Malouf. In unpublished work, Hickerson and Stanley independently proved that the Somos-6 sequence is integer-only. An unpublished proof that Somos-7 is integer-only was found by Ben Lotto in 1990. Fomin and Zelevinsky (2002) gave the first published proof that Somos-6 is integer-only.
However, the 
-Somos
sequences for 
do not give integers. The values of 
for which 
first becomes non-integer for the Somos-
sequence for 
, 9, ... are 17, 19, 20, 22, 24, 27, 28, 30, 33, 34, 36,
39, 41, 42, 44, 46, 48, 51, 52, 55, 56, 58, 60, ... (OEIS A030127).
