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 clusters 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. If you have any questions, please contact me; this site is still very much a work-in-progress.

A history of Somos sequences (somewhat out of date, as it was written by Michael Somos in 1994, but the bibliography gives lots of good places to start one's reading).

An 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). This document is a work in progress, so suggestions are welcome!

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 re-do the theory of elliptic functions in purely combinatorial terms.

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). Also: A slide-show written by Rachel Shipsey that gives a nice introduction to elliptic divisibility sequences (which are intimately connected with Somos sequences); and, the famous Cremona tables of elliptic curves of small conductor. The definitive on-line resource for elliptic divisibility sequences is is Graham Everest's elliptic divisibility sequence web-page.

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.

Somewhat further afield:

Pages relating to Eric Kuo's graphical condensation algorithm: its application to domino tilings of Aztec diamonds, its application to rhombus tilings of hexagons, and the (still mysterious) resemblance between Kuo condensation and Dodgson condensation. See also Kuo's preprint.

An exposition of Dodgson condensation, with an explanation of how both number-walls and frieze-patterns can be viewed as special cases; an account of "antifrieze patterns" with general boundary and their combinatorial significance (an amalgam of postings by Jim Propp, Rick Kenyon, Ira Gessel, Eric Kuo, Robin Chapman, Ron Adin, and Christian Krattenthaler); and an essay on the combinatorics of domino-tilings that takes antifrieze-patterns as a starting-point.

Prominently missing from this site is information on the theory of integrable systems, the Hirota equation, etc. You might check out the article A survey of Hirota's difference equations by Anton Zabrodin.

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, and interpretations of Somos-6 and Somos-7 should not be long in coming. For a preview, see the 2002 REACH tee-shirt.

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).