Difference between methods of converting from finite state machine to regular expression
There is a reference question on methods to convert a finite automaton into a regular expression. Personally I am very fond of the "State removal method" which is equivalent to the "Transitive closure method" you seem to be using. The reason is that the administration you do in the state removal seems to be less, and is more easily kept as labels to edges, rather than keeping track of many $R[i,j,k]$ sets.

Nevertheless, basically all methods use the same recursion, and one gets equivalent results. I say equivalent here, because there is no way to uniquely denote regular languages. Even when using the same method, when applying the reductions in a different order one gets a different result. But again, this is always the same language, only written in a different way.