A Deceptively Simple Quadratic Recurrence

Abstract

Standard techniques for treating linear recurrences no longer apply for quadratic recurrences. It is not hard to determine asymptotics for a specific parametrized model over a wide domain of values (all $p \neq 1/2$ here). The gap between theory and experimentation seems insurmountable, however, at a single outlier ($p = 1/2$).