You don't define your pay periods accurately enough to be certain of the answer.
Superficially, one might assume that If your pay periods are 1-26, to obtain the start date of the pay period, merely subtract 1, multiply by 14, and add that to the first of the year. In other words, assume each pay period is 2 weeks, and that there are 52 weeks in a year.
For example:
(year of interest)
A1: 2013
A2: PayPeriod (your dropdown showing 1-26)
Payperiod start date:
=(a2-1)*14+date(a1,1,1)
HOWEVER, as 26 pay periods in a year implies a 52 week year. 52 weeks is only 364 days. How do you want to treat the extra day(s) each year?
For 2013 the 26th pay period would start on 17 December 2013 and, if it is a 14 day period, end on 30 December 2013. If the payperiods start over in 2014 on 1 January, what happens to 31 December???
Do you want to continue the "every two weeks" in perpetuity from 1/1/2013, or do you need to better define your pay periods?