For each month If you calculate in a non-leap year, the number of days preceding it a given year mod 7, you get

J...0 (number of days preceding JAN = 0)
F..3 (number of days preceding FEB= 31 ie 31 MOD (7) is 3)
M..3 (number of days preceding MARCH  = 59 ie 59 MOD (7 )is 3)
A..6
M..1
J...4
J...6
A..2
S..5
O..0
N..3
D..5

If you apply the same logic to a leap year, you will get the following result

J...0
F..3
M..4
A..0
M..2
J...5
J...0
A..3
S..6
O..1
N..4
D..6

In each case, every number (as there are 7 days in a week, assign a number to each day of the week),  from 0 through 6 occurs at least once every year. That means that, regardless of what day the year starts on, every day of the week will be represented as starting off a month at least once that year. The same is true for other days of the month, such as the 13th. The 13th of the month will fall on each of the 7 days of the week at least once every year.

We can see more from the above tables. It is possible to have just one Friday the 13th, or as many as three in one year.

If a month is going to have a Friday the 13th, it has to start on a Sunday. This will happen three times in a leap year if the year starts on a Sunday. The three months will be January, April, and July, where the three 0's are in the leap year chart.

For non-leap years, we get Fridays on the 13th in February, March, and November whenever the year starts on a Thursday. These are where the 3's are in the first chart.

The numbers that appear just once in each chart, the 1, 2, and 4 in the first chart and the 1, 2, and 5 in the second, mark months that can have a lone Friday the 13th.