shelled and riddled

Consider a shell game with three shells and a ball where only the location of the shell with the ball is exchanged with the location of an empty shell, randomly chosen. If one starts with the ball as rightmost, what is the distribution of the location of the ball after N steps?

Running an exploratory R code like

o=rep(0,3)
for(n in 1:1e6){
  b=c(0,0,1)
  for(t in 1:N){
    i=sample((1:3)[!b],1);b=0*b;b[i]=1}
  o=o+b}

shows that the difference in probability is between the rightmost position and both others, starting at zero, and evolving as p⁺=(1-p⁻)/2, with the successive values 0,1/2,1/4,3/8,5/15,11/32,… Very quickly converging to 1/3.