We came up with this statistic using our historical chart of El Niño (warm phase) and La Niña (cold phase) events here:

www.cpc.ncep.noaa.gov/products/analysis_monitoring/ensostuff/ONI_v5.php

A "first year" La Nina means that there was no La Nina prior to its start.  A "second year" La Nina means that there was a La Nina winter (events tend to reach their strongest point in the winter months) prior to the second winter of La Nina.  So by that count, here's the first year La Nina winters that were followed by La Nina the next winter ( 8 events total ):

1954-55 followed by 1955-56

1970-71 followed by 1971-72 

1973-74 followed by 1974-75

1983-84 followed by 1984-85

1998-99 followed by 1999-2000

2007-08 followed by 2008-09

2010-11 followed by 2011-12

2016-17 followed by 2017-18