Rosh Hashannah (first day) can never occur on Sunday, Wednesday, or Friday so that Yom Kippur cannot occur on Friday or Sunday and Hoshannah Rabbah can never occur on Shabbos.
Judaism 101 gives an interesting mnemonic for determining the leap years in the cycle using a piano keyboard
This means that the pattern for these 5 years are L R R L R
A more detailed explanation is here
This makes the regular year have 353, 354, and 355 days ("adding 3, 4, or 5 days to the day of the week of Rosh Hashannah), while leap years have 383, 384, and 385 days ("adding" 5, 6, and 0). The Jewish year uses a 19 year cycle and the leap years can be shown by taking the year number as modulus 19. This year (5771) is year 14 of the cycle (year 19 of the cycle has modulus 0). The five years involved are thus 14, 15, 16, 17, 18 Year "17" is the next leap year. This year (a leap year) both Cheshvan and Kislev are 30 days so that next Rosh Hashannah (5772) will be also be on Thursday and Friday. The following year will have Cheshvan 29 days and Kislev 30 days (30 Kislev 5772 is 26 December 2011). This brings the following Rosh Hashannah (5773) to Monday and Tuesday. .
5773 will have 29 days in Cheshvan and Kislev which brings Rosh Hashannah back to Thursday again (from the modulus calculation above.) for 5774.
5774 is year 17 of the 19 year cycle and is again a leap year and again will have both Cheshvan and Kislev set to 30 days. This means that the following (regular) year of 5775 will again start on Thursday.
Since 5775 is not a leap year, Rosh Hashannah of 5776 will occur on Monday and since it is again a leap year (modulus number 0) with both Cheshvan and Kislev 30 days, the following year (5777) is again on Monday.
Larry Nussbaum pointed out by email the following five years starting with 5776 are L R R L R as well with the leap year again being "full" (both Cheshvan and Kislev having 30 days) This means that four out of those five years start on Monday, with the "off year" being Thursday. The last year of this pattern (5780) is year number 4 of the cycle, so 5781 (year 5), which starts on Shabbos, is also regular and 5782 starts on Tuesday.
This double pattern is quite rare. This appears to be to only such pattern (2 sets of 4 out of 5 back to back) in the past 247 years. In 5760 - 5764 we had 4 of 5 years with RH on Shabbos. There is no short cycle (like 19 or 28 years) when it comes to days of the week for yom tovim. The only cycle is a 247 year cycle (and that's not 100% guaranteed). Mr. Nussbaum also pointed out that the Tur has a 247 year calendar.
I saw this on the Mail Jewish mailing list, from Richard Fiedler
Mail.Jewish Mailing List
Volume 59 Number 26
Produced: Tue, 14 Sep 2010 01:15:16 EDT
From: Richard Fiedler
Date: Mon, Sep 13,2010 at 10:01 PM
Subject: Dates of Rosh HaShanah
Rosh HaShanah 2013 has another extremely unusual aspect. The Old Moon will be visible on Wednesday Morning Sept 4, 2013 and we will welcome in Rosh HaShanah Wednesday Night.
The Molad of Tishrei 2013 will be Yom Chamishi (Thursday), 16 hours 830 parts. This is under the window of the Molad Zaqen rule (18 hours causes a deferral of Rosh HaShanah) that normally prevents as Old Moon from being seen Erev Rosh HaShanah.
If you will join me in Jerusalem that week I will be on Mount Scopus at 5:10 AM when the Moon will rise. The Sun will rise at 6:16 AM. We will be seeing the cause of the Gemora of Rabban Gamliel and Rabbi Yehoshua (120 CE).
The reason for this is rooted in a very exceptional lunar orbit which is taking a path below the African Continent. Here is a link to the R H van Gent chart of Lunar Visibility for that date. https://files.me.com/richardfiedler/tfsa4d
The above comment shows the "old moon" being visible the day before Rosh Hashanah by the fixed calendar. I realized as a result of a comment that I received that the Sanhedrin is based on eidus, which means that the eidim had to have seen the new moon. IIRC this is a minimum of six hours after the actual astronomic molad, which oscillates around the average announced molad. Actually, a member of the Israel New Moon Society has told me that the new moon would not be visible to the naked eye until it is at least 18 -20 hours old. The question would be when the actual astronomic "new moon" would be. I looked up on the internet and went to a planetary body calendar website to get these values. Note that the current rule is that Daylight time in Israel ends the Sunday morning before Yom Kippur. As a result, DST ends September 8. The times shown below are therefore in Daylight Savings Time.
Moonrise and moonset in Jerusalem (DST)
Sep 4 Sep 5
Time: 11:42 A.M. 12:27 P.M.
Altitude: 65.6° 61.4°
Distance: 396,157 km 392,621 km.
Illuminated: 1.5% 0.1%
Phase: New at 2:37 P.M. Sep 5
I also checked and sunrise is 6:16 and 6:17 A.M. and sunset is 6:59 P.M. and 6:58 PM on those days. This seems to imply that IF the six hour time is correct, then the new moon would probably not be visible until 8:43 P.M. which is well after moonset. Note that the .1% illumination is at the meridian (which is "drowned out" by the noon sun) If the two hours before the new moon are linear to the 4 hours after the new moon, then the illumination would be no more than .2% and the six hours would be about .3% The previous days meridian shows 4.8% illumination which would make the moonrise illumination about 2.5%, with the sun still below the horizon. This may indeed allow it to be seen. In the case of the new moon, since the moon sets 15 minutes earlier than the sun, it would seem to be very difficult to see it. The illumination at the meridian for the day after the new moon (approximately 22 hours) is 1.0%. If the 18 hour estimate mentioned earlier is correct, this implies that the old moon would indeed have been visible.
When the Sanhedrin is reestablished, there will be a set of 30 day months until Rosh Chodesh gets co-ordinated with the eidus and astronomical molad rather than the fixed average molad that we announce now. As a result the Sanhedrin would have had to be established a number of months before this so that Rosh Hashannah could be on Shabbos (September 7) while Elul remained 29 days.