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Somos Sequence

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The Somos sequences are a set of related symmetrical recurrence relations which, surprisingly, always give integers. The Somos sequence of order k, or Somos-k sequence, is defined by

a_n=(sum_(j=1)^(|_k/2_|)a_(n-j)a_(n-(k-j)))/(a_(n-k)),
(1)

where |_x_| is the floor function and a_j=1 for j=0, ..., k-1.

The 2- and 3-Somos sequences consist entirely of 1s. The k-Somos sequences for k=4, 5, 6, and 7 are

a_n=(a_(n-1)a_(n-3)+a_(n-2)^2)/(a_(n-4))
(2)
a_n=(a_(n-1)a_(n-4)+a_(n-2)a_(n-3))/(a_(n-5))
(3)
a_n=1/(a_(n-6))[a_(n-1)a_(n-5)+a_(n-2)a_(n-4)+a_(n-3)^2]
(4)
a_n=1/(a_(n-7))[a_(n-1)a_(n-6)+a_(n-2)a_(n-5)+a_(n-3)a_(n-4)].
(5)

The first few terms are summarized in the following table.

kOEISa_0^((k)), a_1^((k)), ...
4A0067201, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, ...
5A0067211, 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, ...
6A0067221, 1, 1, 1, 1, 1, 3, 5, 9, 23, 75, 421, 1103, ...
7A0067231, 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 In unpublished work, Hickerson and Stanley independently proved that the Somos-6 sequence is integer-only. An unpublished proof that Somos-7 is integer-only was found by Ben Lotto in 1990.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 k-Somos sequences for k>=8 do not give integers. The values of n for which a_n first becomes non-integer for the Somos-k sequence for k=8, 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).

See also

Göbel's Sequence, Heronian Triangle

Portions of this entry contributed by Jim Propp

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References

Buchholz, R. H. and Rathbun, R. L. "An Infinite Set of Heron Triangles with Two Rational Medians." Amer. Math. Monthly 104, 107-115, 1997.Carroll, G. D. and Speyer, D. "The Cube Recurrence." 24 Mar 2004. http://www.arxiv.org/abs/math.CO/0403417/.Fomin, S. and Zelevinsky, A. "The Laurent Phenomenon." Adv. Appl. Math. 28, 19-44, 2002.Gale, D. "Mathematical Entertainments: The Strange and Surprising Saga of the Somos Sequences." Math. Intel. 13, 40-42, 1991.Gima, H.; Matsusaka, T.; Miyazaki, T.; and Yara, S. "On Integrality and Asymptotic Behavior of the (k,l)-Göbel Sequences." 14 Feb 2024. https://arxiv.org/pdf/2402.09064.Malouf, J. L. "An Integer Sequence from a Rational Recursion." Disc. Math. 110, 257-261, 1992.Propp, J. "The Somos Sequence Site." http://jamespropp.org/somos.html.Robinson, R. M. "Periodicity of Somos Sequences." Proc. Amer. Math. Soc. 116, 613-619, 1992.Sloane, N. J. A. Sequences A006720/M0857, A006721/M0735, A006722/M2457, A006723/M2456, and A030127 in "The On-Line Encyclopedia of Integer Sequences."Speyer, D. "Perfect Matchings and the Octahedron Recurrence." 2 Mar 2004. http://www.arxiv.org/abs/math.CO/0402452/.Stone, A. "The Astonishing Behavior of Recursive Sequences." Quanta. Nov. 16, 2023. https://www.quantamagazine.org/the-astonishing-behavior-of-recursive-sequences-20231116.

Referenced on Wolfram|Alpha

Somos Sequence

Cite this as:

Propp, Jim and Weisstein, Eric W. "Somos Sequence." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SomosSequence.html

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