A Deceptively Simple Quadratic Recurrence

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 $p\mathrm{\ne }1\mathrm{/}2$ here). The gap between theory and experimentation seems insurmountable, however, at a single outlier ($p=1\mathrm{/}2$).