Coming home I whipped up a python scriplet that basically did that for the first million digits and found that none of them were divisible by 3 (they weren't divisible by 7 either but I figured I would stick with 3 first). Now we know that n^2 is sometimes divisible by 3 (9, 36, 81 all being examples) and we know that a series that is always divisible by 3 is 3m (two series never divisible by three are 3m+1 and 3m+2). So from this I infer that the series n^2+1 only falls inside of the series 3m+1 and 3m+2 and never on 3m... and n^2 falls on 3m or 3m+1 (since if it fell on 3m+2 then n^2+1 would push those values to 3m). Ok but why?

Looking over the series I came up with the following:

n 1 2 3 4 5 6 7 8 9 10

n^2 1 4 9 16 25 36 49 64 81 100

m{3m+1/3m} 0 1 3 5 8 12 16 21 27 33

diff between m 1 2 2 3 4 4 5 6 6

Ok now I just have more silly numbers and probably a wild goose chase but it is interesting that the series of 3m/3m+1 growth grows in a regular pattern. Anyway, just an odd thing non-fedora related (i hope).