= Solution
The <Parseval identity> for the orthonormal stationary-state expansion is $\sum_{n\geq1}|c_n|^2=\|\psi\|^2=1$. Splitting the amplitudes into even and odd indices gives
$$
1=\frac3{\pi^2}\sum_{p=1}^\infty\frac1{p^2}+\frac{12}{\pi^2}\sum_{p=1}^\infty\frac1{(2p-1)^2}\left[1+\frac{2(-1)^p}{(2p-1)\pi}\right]^2.
$$
Multiply by $\pi^2/3$ to obtain the requested identity. Its integer constants are
$$
\boxed{A=1,\qquad B=4,\qquad C=2.}
$$
The series converges absolutely, since both parity contributions are $O(p^{-2})$. This derivation uses the normalization of the prepared state and completeness of the well <stationary states>, rather than a separate summation trick.
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