An $n\times n$ pentadiagonal matrix (with $n\geq 3$) is a matrix of the form
 $\begin{pmatrix}c_{1}&d_{1}&e_{1}&0&\cdots&\cdots&0\\ b_{1}&c_{2}&d_{2}&e_{2}&\ddots&&\vdots\\ a_{1}&b_{2}&\ddots&\ddots&\ddots&\ddots&\vdots\\ 0&a_{2}&\ddots&\ddots&\ddots&e_{n-3}&0\\ \vdots&\ddots&\ddots&\ddots&\ddots&d_{n-2}&e_{n-2}\\ \vdots&&\ddots&a_{n-3}&b_{n-2}&c_{n-1}&d_{n-1}\\ 0&\cdots&\cdots&0&a_{n-2}&b_{n-1}&c_{n}\end{pmatrix}.$
It follows that a pentadiagonal matrix is determined by five vectors: one $n$-vector $c=(c_{1},\ldots,c_{n})$, two $(n-1)$-vectors $b=(b_{1},\ldots,b_{n-1})$ and $d=(d_{1},\ldots,d_{n-1})$, and two $(n-2)$-vectors $a=(a_{1},\ldots,a_{n-2})$ and $e=(e_{1},\ldots,e_{n-2})$. It follows that a pentadiagonal matrix is completely determined by $n+2(n-1)+2(n-2)=5n-6$ scalars.