= One-to-three oscillator degeneracy
{title2=$E_N=(N+2)\hbar\omega,\quad g_N=\lfloor N/3\rfloor+1$}
For two independent <quantum harmonic oscillators> of frequencies $\omega$ and $3\omega$, the product state with occupation numbers $n_+,n_-\geq0$ has energy $(n_++3n_-+2)\hbar\omega$. For a fixed excitation integer $N$, every allowed pair is $n_-=j$, $n_+=N-3j$ with $0\leq j\leq\lfloor N/3\rfloor$. Each contributes one orthogonal product eigenvector, proving the displayed degeneracy. The offset two is the sum of the two zero-point energies in units of $\hbar\omega$.
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