On Conwayʹs recursive sequence Original Research Article

The recurrence a(n) = a(a(n − 1)) + a(n − a(n − 1)), a(1) = a(2) = 1 defines an integer sequence, publicized by Conway and Mallows, with amazing combinatorial properties that cry out for explanation. We take a step towards unravelling this mystery by showing that a (n) can (and should) be viewed as a simple ‘compression’ operation on finite sets. This gives a combinatorial characterization of a(n) from which one can read off most of its properties. We prove a conjecture of Mallows on the asymptotic shape of a(n), and give an algorithm for solving Conwayʹs challenge problem about the epsilontics of (a(n)/n −1/2). Along the way we encounter some beautiful constructions involving trees, recursively expanding finite strings, and refinements of Pascalʹs triangle. Newmanʹs generalizations of a(n) can be analyzed in the same way, and the results obtained point to possible relations with Conwayʹs theory of games.