Do you mean an+1=(an4+1)/3?
Let's write out a few terms of the three series:
(1) a1=0: 0, 1/3, 10/27, 82/243, ...
(2) a1=1: 1, 2/3, 97/243, ...
(3) a1=2: 2, 17/3, 83602/243, ...
It seems from observation that (3) definitely doesn't converge, while (1) and (2) could converge.
Consider (x4+1)/3<1, then x4+1<3, x4<2, x<1.19 (approx). So a1 must be less than 1.19 for the series to converge, that is, for the nth term to converge to a finite value.
(1) and (2) satisfy this requirement, but (3) does not.