  This present paper is devoted to the study of a class of Nakayama algebras
$N_n(r)$ given by the path algebra of the equioriented quiver $\mathbb{A}_n$
subject to the nilpotency degree $r$ for each sequence of $r$ consecutive
arrows. We show that the Nakayama algebras $N_n(r)$ for certain pairs $(n,r)$
can be realized as endomorphism algebras of tilting objects in the bounded
derived category of coherent sheaves over a weighted projective line, or in its
stable category of vector bundles. Moreover, we classify all the Nakayama
algebras $N_n(r)$ of Fuchsian type, that is, derived equivalent to the bounded
derived categories of extended canonical algebras. We also provide a new way to
prove the classification result on Nakayama algebras of piecewise hereditary
type, which have been done by Happel--Seidel before.
