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