  We revisit the one dimensional discrete time quantum walk with 3 states and
the Grover coin. We derive analytic expressions for observed the localization,
an long time approximation for the probability density function (PDF). We also
connect the time averaged approximation to the PDF found by Inui et. al. to a
spatial average of the walk. We show that this smooth approximation constitutes
a proper PDF that predicts moments of the real PDF accurately.
