This is a Markov chain in state-space $\mathbb R^n$. There is machinery to determine whether (and to what) it converges. You determine that a certain $n \times n$ matrix is "irreducible" and then you get convergence to the unique (up to scalar multiple) positive eigenvector with eigenvalue $1$. Goes back to Perron & Frobenius, I guess.
Maybe this 3-term average has a name in probability theory (but I don't know one) ... However I really doubt is has a name in number theory.