I know that there exist a closed form solution for the eigenvalues of tridiagonal Toeplitz matrix but how about a symmetric pentadiagonal Toepltz matrix.
plz look at
http://www.sciencedirect.com/science/article/pii/S0024379511004307
https://repositorium.sdum.uminho.pt/bitstream/1822/11391/1/RSBT_final.pdf
anoher good refrence
skew-symetric case
http://www.sciencedirect.com/science/article/pii/S0096300311011325
http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=165639&fileId=S0962492902000016
U. Grenander, G. Szegó
Toeplitz Forms and their Applications
Univ. of California Press (1958)
another good reference is
http://iopscience.iop.org/article/10.1088/0266-5611/21/6/005/meta
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