Let $n$ be even and consider the non-normalized discrete Sine transform of type 5 which is

$$S=\left(\sin(k+1)(l+1)\frac{\pi}{n+\frac12}\right)\_{k,l=0}^{n-1}$$

Let us denote $s\_{-,l}$ by the $l^{th}$-column of $S$. It can be considered as a $n$-tuple in $\mathbb{R}^n$.

**Q.** I am looking for $n$-tuples $v=(v\_0,\cdots,v\_{n-1})$ in $\mathbb{R}^n$ satisfying the following conditions:

1- $v\_j=\sin2(j+1)\frac{\pi}{n+\frac12}$ if $j$ is even.

2- The following are valid concerning inner products: $$\langle v , s\_{-,l} \rangle=\left\{
 \begin{array}{cl}
 1 & l=0 \\
 0 & l\neq 0~,~ l \operatorname{is even}
 \end{array}
\right.$$