From a thoughtlessness point of view, it is not hard to use a computer simulation to find out the location of the sole survivor for a given n, until n gets to around 10^8 depending on the computer. Obviously since the algorithm is deterministic there is always a unique sole survivor.
I somewhat cheated. If we consider starting on #1 instead of #2 (which is how we might expect the problem to be stated in "standard form"), then it is easy to work out on paper the start of the sequence, and using OEIS quickly reveals A152423. This should be more than enough of a head start for you.