Signature of a tridiagonal matrix

Question

Let $A$ be a tridiagonal matrix of the form $$A= \begin{pmatrix} a_1 & 1 \\ 1 & a_2 & 1 \\ & 1 & \ddots & \ddots \\ & & \ddots & \ddots & 1 \\ & & & 1 & a_n \end{pmatrix}, $$ where the diagonal elements $a_i$ are integers. Since $A$ is symmetric, it has $n$ real eigenvalues (up to algebraic multiplicity) $\lambda_1,\dots,\lambda_n\in\mathbb{R}$. We define the signature of $A$ as the number of positive eigenvalues minus the number of negative eigenvalues. In this particular case, does there exist a simpler expression for the signature of $A$ in terms of the integers $a_i$? I know Sylvester's law of inertia can be used if we manage to find the appropriate similarity transformation, but I'm not sure what kind of similarity transformation would make matters simpler in this case.