Extended near Skolem sequences, Part III

A $k$-extended $q$-near Skolem sequence of order $n$, denoted by $N_n^q\left(k\right)$, is a sequence $s_1,s_2,\text{…},s_{2n-1}$ where $s_k=0$ and for each integer $\ell \in \left[1,n\right]\backslash \left\{q\right\}$ there are two indices $i$, $j$ such that $s_i=s_j=\ell$ and $\mid i-j\mid =\ell$. For an $N_n^q\left(k\right)$ to exist it is necessary that $q\equiv k\left(\mathrm{mod}2\right)$ when $n\equiv 0,1\left(\mathrm{mod}4\right)$ and $q\not\equiv k\left(\mathrm{mod}2\right)$ when $n\equiv 2,3\left(\mathrm{mod}4\right)$, where $(n,q,k)\ne (3,2,3)$, $\left(4,2,4\right)$. Any triple $\left(n,q,k\right)$ satisfying these conditions is called admissible. In this manuscript, which is Part III of three manuscripts, we construct the remaining sequences; that is, $N_n^q\left(k\right)$ for all admissible $(n,q,k)$ with $q\in \left(\lfloor \frac{n+2}{3}\rfloor ,\lfloor \frac{n-2}{2}\rfloor \right)$ and $k\in \left(\lfloor \frac{2n}{3}\rfloor ,n-1\right)$.