A directed Toeplitz graph T n s 1 , ⋯ , s k ; t 1 , ⋯ , t l with vertices 1 , 2 , ⋯ , n is a directed graph whose adjacency matrix is a Toeplitz matrix. In this paper, we investigate the Hamiltonicity in directed Toeplitz graphs T n s 1 , ⋯ , s k ; t 1 , ⋯ , t l with s 1 = 1 and s 3 = 4.