We prove the existence and uniqueness of a weighted pseudo asymptotically mild solution to the following class of abstract semilinear difference equations: $$ u(n+1)= A \sum_{k=-\infty }^{n} a(n-k)u(k+1)+ \sum _{k=-\infty }^{n} b(n-k)f\bigl(k,u(k)\bigr),\quad n\in \mathbb{Z}, $$ where A is the generator of a resolvent sequence $\{S(n)\}_{n\in \mathbb{N}_{0}}$ of bounded and linear operators defined in a Banach space X, the sequences $a, b$ are complex-valued, and $f\in l^{1}( \mathbb{Z}\times X, X)$ .