Using the GNU Date:
$ d=; n=0; until [ "$d" = "$enddate" ]; do ((n++)); d=$(date -d "$startdate + $n days" +%Y%m%d); echo $d; done
20160513
20160514
Or, a few lines:
startdate=20160512
enddate=20160514
d=
n=0
until [ "$d" = "$enddate" ]
do
((n++))
d=$(date -d "$startdate + $n days" +%Y%m%d)
echo $d
done
How it works
d=; n=0
Initialize variables.
until [ "$d" = "$enddate" ]; do
Run a loop that ends on enddate.
((n++))
Increase the day counter.
d=$(date -d "$startdate + $n days" +%Y%m%d)
Calculate the date ndays after startdate.
echo $d
Display date.
done
End of cycle signal.