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A $ n \times n $ matrix $A$ is a tridiagonal symmetric marix such that its diagonal entries are all 2 except the final $n \times n$ element which is 1 and its superdiagonal and subdiagonal elements are all -1.

Is there any analytic method to find its eigenvalues?

I know how to find the eigenvalues of this type matrix when the last element is alos 2 by solving corresponding linear recurrence euation as in

But the same method seems to be fail in my case.

Thank you in advance.

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