Extended near Skolem sequences, Part III

A k $k$ ‐extended q $q$ ‐near Skolem sequence of order n $n$ , denoted by Nnq(k) ${{\mathscr{N}}}_{n}^{q}(k)$ , is a sequence s1,s2,…,s2n−1 ${s}_{1},{s}_{2},\ldots ,{s}_{2n-1}$ where sk=0 ${s}_{k}=0$ and for each integer ℓ∈[1,n]\{q} $\ell \in [1,n]\backslash \{q\}$ there are two indices i $i$ , j $j$ such that si=sj=ℓ ${s}_{i}={s}_{j}=\ell $ and ∣i−j∣=ℓ $| i-j| =\ell $ . For an Nnq(k) ${{\mathscr{N}}}_{n}^{q}(k)$ to exist it is necessary that q≡k(mod2) $q\equiv k\,(\mathrm{mod}\,2)$ when n≡0,1(mod4) $n\equiv 0,1\,(\mathrm{mod}\,4)$ and q≢k(mod2) $q\not\equiv k\,(\mathrm{mod}\,2)$ when n≡2,3(mod4) $n\equiv 2,3\,(\mathrm{mod}\,4)$ , where (n,q,k)≠(3,2,3) $(n,q,k)\ne (3,2,3)$ , (4,2,4) $(4,2,4)$ . Any triple (n,q,k) $(n,q,k)$ satisfying these conditions is called admissible. In this manuscript, which is Part III of three manuscripts, we construct the remaining sequences; that is, Nnq(k) ${{\mathscr{N}}}_{n}^{q}(k)$ for all admissible (n,q,k) $(n,q,k)$ with q∈⌊n+23⌋,⌊n−22⌋ $q\in \left[\lfloor \frac{n+2}{3}\rfloor ,\lfloor \frac{n-2}{2}\rfloor \right]$ and k∈⌊2n3⌋,n−1 $k\in \left[\lfloor \frac{2n}{3}\rfloor ,n-1\right]$ .