Somos Sequence

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)
			(4)
			(5)

The first few terms are summarized in the following table.

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 In unpublished work, Hickerson and Stanley independently proved that the Somos-6 sequence is integer-only. An unpublished proof that Somos-7 is integer-only was found by Ben Lotto in 1990.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

Wolfram Web Resources