One-to-three oscillator degeneracy ($E_N=(N+2)\hslash \omega ,g_N=\lfloor N/3\rfloor +1$)

For two independent quantum harmonic oscillators of frequencies $\omega$ and $3\omega$, the product state with occupation numbers $n_{+},n_{-}\ge 0$ has energy $(n_{+}+3n_{-}+2)\hslash \omega$. For a fixed excitation integer $N$, every allowed pair is $n_{-}=j$, $n_{+}=N-3j$ with $0\le j\le \lfloor N/3\rfloor$. 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 $\hslash \omega$.