Inverse of tridiagonal Toeplitz matrix has no zero entries

Question

The inverse of the symmetric tridiagonal matrix (Toeplitz) $$ t_{ij}=\begin{align} \begin{cases} -2 &\quad \text{if} \,\, i=j \\ 1 &\quad\text{if} \,\, \vert i-j\vert = 1 \end{cases} \end{align} $$

does not have any non zero entries according to online inverse calculators (I tried up to $5 \times 5-$matrices). Why is it so?

Answer 1

The expression for the $i,j$ element of the inverse of this $n\times n$ matrix has a nice form \begin{align} w_{ij}&=\frac{i\,j}{n+1}-\min(i,\,j) \tag{1}\label{1} , \end{align}

so $w_{ij}$ could be zero only if \begin{align} n&=\frac{i\,j}{\min(i,\,j)}-1 =\max(i,j)-1=n-1 , \end{align}

and the answer follows.