Density of free particle energy eigenfunctions in 1d for periodic boundary conditions

The energy spectrum for the above, subject to the periodic boundary conditions such that the eigenfunctions are have a period of $L$, is given by
$$E_n = E_0n^2, qquad n=0, pm1, pm2, ldots$$
where $E_0 = (2pi^2hbar^2)/(mL^2)$. Each $Eneq0$ is doubly degenerate.

Then to find the density of states, I observe that $E_{n+1}-E_n = E_0(2n+1)approx 2E_0n=2sqrt{E_0E_n}$. Since there’s double degeneracy, I conclude that the energy density at energy $E$ is
$$frac{2}{2sqrt{E_0E}}=sqrt{frac{m}{2E}}frac{L}{pihbar}.$$

But Merzbacher in his Quantum Mechanics says that it should be
$$sqrt{frac{2m}{E}}frac{L}{pihbar}.$$

Where did I go wrong?