**Abstract:** We study the $T$-system of type $A_\infty$, also known as the octahedron recurrence/equation, viewed as a $2+1$-dimensional discrete evolution equation. The solution is expressed algebraically in terms of the stepped surface over which the initial data are specified as "slanted". Last semester, we have discussed the "solution" of the $T$-system with this initial data as Gale-Robinson sequence, where the value is uniform (with respect to some structure of the stepped surface). In this talk, I will discuss the solution where the value of initial data possessing a $2\times 2$-periodicity.