Tridiagonal matrix

Definition:
A square matrix with nonzero elements only on the diagonal and along the subdiagonal and superdiagonal.

A simple example: $$\begin{vmatrix}
2 & -1 & & & & \\
-1 & 2 & -1 & & & \\
& -1 & 2 & -1 & & \\
& & -1 & 2 & -1 &\\
& & & -1 & 2 & -1 \\
& & & & -1 & 2 \\
\end{vmatrix}$$ We can make use of recurrence relation to find this determinant.
Expand the determinant by the first row: $$2\begin{vmatrix}
2 & -1 & & & \\
-1 & 2 & -1 & & \\
& -1 & 2 & -1 &\\
& & -1 & 2 & -1 \\
& & & -1 & 2 \\
\end{vmatrix}+
\begin{vmatrix}
-1 &  -1 & & & \\
& 2 & -1 & & \\
& -1 & 2 & -1 &\\
& & -1 & 2 & -1 \\
& & & -1 & 2 \\
\end{vmatrix}$$ which is equal to $$2\begin{vmatrix}
2 & -1 & & & \\
-1 & 2 & -1 & & \\
& -1 & 2 & -1 &\\
& & -1 & 2 & -1 \\
& & & -1 & 2 \\
\end{vmatrix}-
\begin{vmatrix}
2 & -1 & & \\
-1 & 2 & -1 &\\
& -1 & 2 & -1 \\
& & -1 & 2 \\
\end{vmatrix}$$ Let $f_n$ be the determinant of the $n\times n$ tridiagonal matrix with diagonal elements all equal to $2$, the sub and super diagonal elements all $-1$.

In general, we have $$f_n=2f_{n-1}-f_{n-2}\\ f_1=2, f_2=3, f_n=n+1.$$ Thus, the determinant of this $6\times 6$ matrix is $\fbox7$.

More to explore:
continuant
continued fraction
http://www.sciencedirect.com/science/article/pii/S0096300307007825
Chapter 6, 7