If you make a table of the korbanot for each holiday, you will notice an interesting pattern:
On each of the Tishrei holidays (except Succot), i.e. Rosh Ha'shana, Yom Kippur, and Shmini Atzeret, we offer an identical korban musaf - 1 "par", 1 "ayil", and 7 "kevasim".
On each of the "shalosh regalim" (except Succot), i.e. Chag ha'Matzot and Shavuot, we offer an identical korban musaf - 2 "parim", 1 "ayil", and 7 "kevasim".
The musaf of Succot is quite different, each day the amount of "parim" changes (from 13 down to 7), and each day we offer 2 "aylim" and 14 "kevasim"! [See Bamidbar 29:12-34.]
The additional "parim" are the most striking difference, and Chazal explain that these are added for the 70 nations for whom Succot is also celebrated (13+12+11+10+9+8+7=70). [See also Zecharya chapter 14, haftara on first day Succot.]
However, the extra "ayil" & "kevasim" also require explanation.
Note, that they are double the number that are offered on all of the other holidays. In other words, instead of 1 "ayil", we bring 2 "aylim"; instead of 7 "kevasim", we bring 14 "kevasim". This indicates that there must be something 'double' about Succot.
The answer may be quite simple. Succot is both one of the "shalosh regalim" (see Shmot 23:14-17 & Devarim 16:1-17), and a Tishrei holiday as well. Therefore, it requires a double musaf. In other words, it should have daily:
|3 "parim" [2+1];|
|2 "aylim" [1+1];|
|14 "kevasim" [7+7].|
However, we add an additional 49 "parim" [10+9+8+7+6+5+4] to reach a total of 70 [49+(3x7)=49+21=70], as explained above.
[Again we find 49 [7x7] related to the shalosh regalim.]
This may reflect the double nature of Succot. On the one hand it is one of the "shalosh regalim" in which we thank Hashem for our harvest of the agricultural year which has just finished. At the same time, we stand in anticipation of the agricultural year which is about to begin, awaiting its important rainy season (see shiur on Rosh Ha'shana), just as we do on all of the other Tishrei holidays.
This 'double nature' is reflected by the two times in daily davening on Succot when we hold the lulav & etrog, during:
|1)||Hallel - to thank God for the harvest of the past year;|
|2)||Hoshanot - to pray to God in anticipation of the new year.|
Similarly, this 'double nature' may also reflect the two reasons that Chazal [see Succah 11b] give us for sitting in the succah.
|1)||Succot mamash - real booths, to protect us from the sun.
This may reflect the aspect of the harvest holiday, where we need to build temporary booths in the field as we gather our fruits in the fields.
|2)||Succot k'neged anannei ha'kavod - representing God's
shchina which protected Bnei Yisrael in the desert.
This may reflect a more spiritual aspect of the holiday, as we attempt to re-live an intimate relationship with Hashem [as we explained in the above shiur].