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Somos Sequence

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The Somos sequences are sequences defined by a family of symmetric rational recurrence relations. For an integer k>=2, the Somos sequence of order k, or Somos-k sequence, is defined for integers n>=k 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 are given by Speyer (2007, p. 329) and for Somos-6 and Somos-7 by Carroll and Speyer (2004, p. 20).

The integer-only property of Somos-4 through Somos-7 is an instance of the Laurent phenomenon. If their initial values are treated as indeterminates, every subsequent term is a Laurent polynomial with integer coefficients in those initial values. Setting all the initial values equal to 1 makes every monomial denominator equal to 1 and therefore produces integers (Fomin and Zelevinsky 2002).

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 In unpublished work, Hickerson and Stanley independently proved that the Somos-6 sequence is integer-only. An unpublished proof that Somos-7 is integer-only was found by Ben Lotto in 1990.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

It is conjectured that the Somos-k sequence eventually contains a noninteger for every k>=8. 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

Gale-Robinson Sequence, Göbel's Sequence, Heronian Triangle, Laurent Phenomenon

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. E. "The Cube Recurrence." Electron. J. Combin. 11, Art. R73, 2004. https://doi.org/10.37236/1826.Fomin, S. and Zelevinsky, A. "The Laurent Phenomenon." Adv. Appl. Math. 28, 119-144, 2002. https://doi.org/10.1006/aama.2001.0770.Gale, D. "Mathematical Entertainments: The Strange and Surprising Saga of the Somos Sequences." Math. Intel. 13, 40-43, 1991. https://doi.org/10.1007/BF03024070.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/abs/2402.09064.Malouf, J. L. "An Integer Sequence from a Rational Recursion." Disc. Math. 110, 257-261, 1992. https://doi.org/10.1016/0012-365X(92)90714-Q.Numberphile. "The Troublemaker Number." May 23, 2022. https://www.youtube.com/watch?v=p-HN_ICaCyM.Propp, J. "The Somos Sequence Site." https://faculty.uml.edu/jpropp/somos.html.Robinson, R. M. "Periodicity of Somos Sequences." Proc. Amer. Math. Soc. 116, 613-619, 1992. https://doi.org/10.1090/S0002-9939-1992-1140672-5.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. E. "Perfect Matchings and the Octahedron Recurrence." J. Algebraic Combin. 25, 309-348, 2007. https://doi.org/10.1007/s10801-006-0039-y.Stone, A. "The Astonishing Behavior of Recursive Sequences." Quanta. Nov. 16, 2023. https://www.quantamagazine.org/the-astonishing-behavior-of-recursive-sequences-20231116/.

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Somos Sequence

Cite this as:

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

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