Without more information I'm making some assumptions with this solution, but here is an off the cuff/rough idea using t-SQL...at least a starting point:
select a.name
,a.date
FROM table a
INNER JOIN table a2 ON a.name = a2.name
WHERE a2.date >= DATEADD(dd,(7 - (DATEPART(dw,a.date)%7)) +1, a.date)
AND a2.date < DATEADD(dd, 7, (DATEADD(dd,(7 - (DATEPART(dw,a.date)%7)) +1, a.date)))
This assumes that you literally want entries that appear in week one and then again in week two where week two is any day from the following Sunday through Saturday (or whichever day you decide to define as the start of the week). I believe this meets your requirement and is a little more specific than defining the following week as simply '>= +7 and <= +14 days'. I'm not saying there is anything wrong with that, just that it might not meet your needs if you want an entry that falls on a Friday and an entry that falls on the following Monday to be considered as a consecutive pair.
This example takes your date, finds the integer representing the day of week it falls on, then mods and subtracts from 7 to find the number of days until the end of that day's week. Then, using that end-of-week date, we can add one day to find the start of the following week as well as use the DATEADD function to add 7 days to find the upper limit (the start of the second week out), giving us a date range to work within. This works well because it is always relevant to the date being passed and you don't have to worry about the position (in the week) of the date being passed in. On the down side, it's probably not going to be a performance champ...so depending on the use and volume, it could be pretty slow.