Wait a second, you don't see a problem with your so called proof? I suck at mathematics, I'll admit to that, but if you add numbers in threes, your answer will be in threes as well.
If you were to do 2 numbers, you'd get answers in twos as well, let's see
11 >> 1 + 1 = 2
22 >> 2 + 2 = 4
etc
What about 5 numbers instead, will it give back answers in fives? Let us find out.
11111 >> 1 + 1 + 1 + 1 + 1 = 5
22222 >> 2 + 2 + 2 + 2 + 2 = 10
Woah, it does. The fact that you picked 3 of something as the starting point means you have a bias for the number 3, but it doesn't make 3 special, because literally every number has the same thing going on.
Even bad at math as I may be, the number 3 has no significant difference from any other number in this regard.
The recurring sequence isn't magnificent, it is plain, and it works for every number.
Edited by Zucadragon, : Typo fixing
™ Version 4.2
