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)
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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)
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(3)
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(4)
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(5)
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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 In unpublished work, Hickerson and Stanley independently proved that the Somos-6 sequence is integer-only. An unpublished proof that Somos-7 is integer-only was found by Ben Lotto in 1990.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).