A leap year starting on Thursday is any year with 366 days (i.e. it includes 29 February) that begins on Thursday, 1 January, and ends on Friday, 31 December. Its dominical letter hence is DC, such as the years 1948, 1976, 2004, 2032, 2060, 2088, and 2128 in the Gregorian calendar or, likewise, 1988 and 2016 in the obsolete Julian calendar.
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Gregorian Calendar
Leap years that start on Thursday occur 13 (or ≈ 13.4%) out of all 97 total leap years in a 400 cycle. Their overall occurrence is 3.25%. For this kind of year, the corresponding ISO year has 53 weeks, and the ISO week 10 (which begins March 1) and all subsequent ISO weeks occur earlier than in all other years. That means, moveable holidays may occur one calendar week later than otherwise possible, e.g. Gregorian Easter Sunday in week 17 about once per leap cycle.
Julian Calendar
Like all leap year types, the one starting with 1 January on a Thursday occurs exactly once in a 28-year cycle in the Julian calendar, i.e. in 3.57% of years.
The final two digits of Julian years repeat after 700 years, i.e. 25 cycles.