The Frobenius dimension $F(A)$ of an Artin algebra $A$ is given by the dimension of $Hom(D(A),A)$ (see the answer in On nearly Frobenius algebras ).
Question 1: Is it true that $F(A) \geq gldim(A)$ for any Nakayama algebra $A$ of finite global dimension?
Question 2: What is the maximal Frobenius dimension of a quasi-hereditary Nakayama algebra with $n$ simple modules? For $n \geq 2$, the sequence starts with 5,10,17,26,37,50,65. It seems like the answer is $n^2+1$ but I do not see an easy reason why such a simple result might be true.
For general Nakayama algebras with finite global dimension the maximal Frobenius dimension seems to be more complicated, at least it does not appear in the oeis.