From the table, we can see that Friday the 13th occurs in two consecutive months in only one case:  The year is not a leap year, and January 1 occurs on a Thursday.  This is case 5 on the table.  By looking at a calendar, we can see that this occurred in 1998.  To see when this will occur again, we need to find the pattern for the years.  For example, if 1998 is case 5, which case is 1999?  2000?

We can figure this out by looking at the first and the last day of the years.  In years that are not leap years, January 1 and December 31 are the same day of the week.  (We can see this by looking at the spreadsheet.  They are both 1 mod 7.)  So for any given year that is not a leap year, the next year starts of the day of the week directly after that day of the week.  For example, since 1998 started on a Thursday and is not a leap year, 1999 will start on a Friday.  1998 is case 5, and 1999 is case 6.

For leap years, it is a little bit different.  By looking at the spreadsheet, we can see that January 1 is 1 mod 7, and December 31 is 2 mod 7.  Thus, the year that occurs after a leap year starts on the day of the week that is two days after the day of the week on which the leap year starts.  For example, 2000 is a leap year and starts on Saturday.  Thus, 2001 starts on a Monday.  2000 is case 14, and 2001 is case 2.

Recall that cases 8 through 14 are the same as cases 1 through 7 except they are leap years.  Cases 1 and 8 both start on Sunday, cases 2 and 9 (2+7) both start on Monday, cases 3 and 10 (3 + 7) both start on Tuesday, etc.

Let's consider a few examples to see what the pattern is:

    1998:    Case 5

    1999:    Previous year is not a leap year, so previous case plus 1:  6

    2000:    Previous year is not a leap year, so previous case plus 1: 7.  
                This year starts on the same day of the week as year 7, but it is a leap year so we add 7.
                Thus, it is case 14.

    2001:    Previous year is a leap year.  
                Previous case minus 7 gives us the day of the week that the previous year started with, 
                and add 2 because the previous year is a leap year.  This case is 14 - 7 + 2 = 2.

    2002:    Previous year is not a leap year, so previous case plus 1: 2 + 1 = 3.

    2003:    Previous year is not a leap year, so previous case plus 1: 3 + 1 = 4.

    2004:   Previous year is not a leap year, so previous case plus 1: 4 + 1 = 5.  
                Add 7 because this year is a leap year.  5 + 7 = 12.

    2005:    Previous year is a leap year.  Previous case - 7 + 2 = 12 -7 + 2 = 7

    2006:    Previous year is not a leap year, so previous case plus 1:  7 + 1 = 8.  
                BUT 8 represents a leap year, but this is not a leap year, so we need 8 mod 7 = 1.

    2100:    Note that is year is not a leap year.

Now that we see the pattern, we can make a spreadsheet to see the pattern for the years 1998 - 2100.  (In the table below, 0 is the same as case 7.)

To see which years will have consecutive Friday the 13th's, we just need to look down the table and see which years are case 5.  They are:

2009, 2015, 2026, 2037, 2043, 2054, 2065, 2071, 2082, 2093, and 2099.