The Somos Sequence Site

Here are some web-resources for information on Somos sequences (invented by Michael Somos) and allied topics. In a few cases I've tried to arrange batches of related articles in a pedagogically suitable order, but this site is very far from being the sort of tutorial one reads linearly from start to finish. Also, I've made no consistent attempt to include articles written after 2010, and I plan to make no further changes to the page in 2024 or beyond. If you have any questions or if any links are broken, please contact me.

A history of Somos sequences (somewhat out of date, as it was written by Michael Somos in 1994, but the bibliography gives lots of good places to start one's reading).

A somewhat dated attempt at a chronology of the Somos-4, Somos-5, Somos-6, and Somos-7 sequences (these being the non-trivial ones whose terms are whole numbers).

Some conjectures relating to Robinson's work on Somos sequences.

A problem (see "Fifth Day") posed by Don Zagier, in which he states an intriguing possibility (which actually was what motivated Somos in the first place): one might introduce (and possibly develop) the theory of elliptic functions in terms of recurrence relations. See Zagier's solution.

David Gale's two articles on Somos sequences are "The Strange and Surprising Saga of the Somos Sequences" (Mathematical Intelligencer 13, no. 1, pp. 40-42 (1991)) and "Somos Sequence Update" (Mathematical Intelligencer 13, no. 4, pp. 49-50 (1991)). The articles are also available in Gale's book "Tracking the Automatic Ant", which collected his Intelligencer articles; see pages 2-5 and 22-24.

An explanation of the connection between a Somos-type sequence and an elliptic curve (by Noam Elkies, with an addendum by David Speyer).

Michael Somos has found an exact formula for the Somos 6 sequence in terms of theta functions. See also his page on the Somos 7 sequence.

Alf van der Poorten's article Elliptic curves and continued fractions, which elaborates on the link between Somos sequences, elliptic curves, and continued fractions. See also his articles Recurrence Relations for Elliptic Sequences: every Somos 4 is a Somos k and Genus 2 curves, continued fractions, and Somos sequences.

Andrew Hone's articles Elliptic Curves and Quadratic Recurrence Sequences, (Bull. Lon, Math. Soc. 37 2 (2005) 161--171) and Sigma function solution of the initial value problem for Somos 5 sequences (Trans. Amer. Math. Soc., 2006) give different algebraic formulas than Elkies', and may be more useful for some purposes. See also the paper Integrality and the Laurent phenomenon for Somos 4 sequences by Christine Swart and Andrew Hone.

Lecture notes for a talk, entitled Somos sequences and bilinear combinatorics, given by James Propp in the Fall of 2000 at the MIT Combinatorics Seminar.

The handout from a talk, entitled Number Walls in Combinatorics, given by Michael Somos in the Fall of 2000 at the MIT Combinatorics Seminar.

The article The Laurent phenomenon, in which authors Sergei Fomin and Andrei Zelevinsky present a general method for proving integrality and Laurentness of the solutions to a wide range of Somos-like recurrences.

Recently, two groups of researchers, one in Canada and France (Mireille Bousquet-Melou and Julian West) and one in the U.S. (my undergraduate research group), working from some of my suggestions about multivariate generalizations of the Somos sequences, have found combinatorial interpretations of Somos-4 and Somos-5. See Perfect Matchings and The Octahedron Recurrence by David Speyer and Perfect matchings for the three-term Gale-Robinson sequences by Mireille Bousquet-Melou, James Propp, and Julian West.

A.V. Ustinov, An Elementary Approach to the Study of Somos Sequences.

A.V. Ustinov, On Somos-4 and Somos-5 Sequences.

V. A. Bykovskii and A. V. Ustinov, On the Laurent Phenomenon for Somos-4 and Somos-5 Sequences.

Andrew Hone, Heron triangles with two rational medians and Somos-5 sequences

For more of Michael Somos' perspective, see his webpage WXYZ Math Project describing polynomial sequence analogues of the Jacobi and Weierstrass elliptic functions.

The Robbins forum (named after David Robbins) is an email forum dedicated to improving our understanding of algebraic recurrences like the David Robbins' octahedron recurrence (which assisted in the discovery of combinatorial interpretations of the Somos sequence).