  We consider a variant on the Tetranacci sequence, where one adds the previous
four terms, then divides the sum by two until the result is odd. We give an
algorithm for constructing "initially division-poor" sequences, where over an
initial segment one divides by two only once for each term. We develop a
probabilistic model that suggests that "most" sequences are unbounded, and
provide computational data to support the underlying assumptions of the model.
