The unreasonable rigidity of Ulam sequences

Abstract

An Ulam sequence U(a,b) is defined as a sequence starting with integers a,b such that 0<a<b, and such that every subsequent term is the smallest integer that can be written as the sum of distinct previous terms in exactly one way. We investigate a new rigidity phenomenon for families U(a,b), where a is fixed and we allow b to increase, centering around a result proved using a novel model theoretic argument. For the specific case U(1,n), we provide an algorithm for computing relevant coefficients, together with a proof of correctness.