  We study a Seshadri constant at a general point on a rational surface whose
anticanonical linear system contains a pencil. First, we describe a Seshadri
constant of an ample line bundle on such a rational surface explicitly by the
numerical data of the ample line bundle. Secondly, we classify log del Pezzo
surfaces which are special in terms of the Seshadri constants of the
anticanonical divisors when the anticanonical degree is between 4 and 9.
