This paper develops a new family of locally recoverable codes for distributed storage systems, Sequential Locally Recoverable Codes (SLRCs) constructed to handle multiple erasures in a sequential recovery approach. We propose a new connection between parallel and sequential recovery, which leads to a general construction of q-ary linear codes with information $(r, t_i, δ)$-sequential-locality where each of the $i$-th information symbols is contained in $t_i$ punctured subcodes with length $(r+δ-1)$ and minimum distance $δ$. We prove that such codes are $(r, t)_q$-SLRC ($t \geq δt_i+1$), which implies that they permit sequential recovery for up to $t$ erasures each one by $r$ other code symbols.