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.