Gale-Robinson Sequence

A Gale-Robinson sequence is a sequence defined by a quadratic rational recurrence equation generalizing the Somos sequences. Its three-term form is

where , , and are distinct positive integers, , and (Fomin and Zelevinsky 2002). The initial values are , , ..., . Gale and Robinson conjectured that all terms are integers when these initial values are 1 and , , and are positive integers (Gale 1991, Fomin and Zelevinsky 2002). Fomin and Zelevinsky (2002) proved the stronger Laurent phenomenon: every term is a Laurent polynomial in the initial values with coefficients in . The conjecture follows by specializing the initial values to 1.

The two-term version has the form

where , , and are positive integers with . It has the same Laurent phenomenon, with coefficients in (Fomin and Zelevinsky 2002, Trott 2006). These statements concern rational functions of independent initial variables. A numerical evaluation of the displayed recurrences requires each divisor to be nonzero.

With , the choices and give Somos-4 and Somos-5, respectively. With , the choices and in the three-term form give Somos-6 and Somos-7.