Let an 8x8 2-tridiagonal Toeplitz matrix is of the form S1. From the literature Eigenvalues of 2-tridiagonal Toeplitz matrix its easy to findout the maximum eigenvalue of S1.
S1=$ \begin{bmatrix} a & 0 & b & 0 & 0 & 0 & 0 & 0 \\ 0 & a & 0 & b & 0 & 0 & 0 & 0 \\ b & 0 & a & 0 & b & 0 & 0 & 0 \\ 0 & b & 0 & a & 0 & b & 0 & 0 \\ 0 & 0 & b & 0 & a & 0 & b & 0 \\ 0 & 0 & 0 & b & 0 & a & 0 & b \\ 0 & 0 & 0 & 0 & b & 0 & a & 0 \\ 0 & 0 & 0 & 0 & 0 & b & 0 & a \end{bmatrix} $
Is there any recurrence relation which can be formulated to find the maximum eigenvalue of variants of the two-tridiagonal Toeplitz matrix, like S2, S3 and S4 given,
S2=$ \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & a & 0 & b & 0 & 0 & 0 \\ 0 & 0 & 0 & a & 0 & b & 0 & 0 \\ 0 & 0 & b & 0 & a & 0 & 0 & 0 \\ 0 & 0 & 0 & b & 0 & a & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} $
s3=$ \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & a & 0 & b & 0 & 0 & 0 & 0 \\ 0 & 0 & a & 0 & b & 0 & 0 & 0 \\ 0 & b & 0 & a & 0 & b & 0 & 0 \\ 0 & 0 & b & 0 & a & 0 & b & 0 \\ 0 & 0 & 0 & b & 0 & a & 0 & 0 \\ 0 & 0 & 0 & 0 & b & 0 & a & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} $
s4=$ \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & a & 0 & b & 0 & 0 & 0 & 0 \\ 0 & 0 & a & 0 & b & 0 & 0 & 0 \\ 0 & b & 0 & a & 0 & b & 0 & 0 \\ 0 & 0 & b & 0 & a & 0 & b & 0 \\ 0 & 0 & 0 & b & 0 & a & 0 & b \\ 0 & 0 & 0 & 0 & b & 0 & a & 0 \\ 0 & 0 & 0 & 0 & 0 & b & 0 & a \end{bmatrix} $