The theory of entanglement-assisted quantum error-correcting codes (EAQECCs) is a generalization of the standard stabilizer quantum error-correcting codes, which can be possibly constructed from any classical codes by relaxing the duality condition and utilizing preshared entanglement between the sender and receiver. In this paper, a new family of EAQECCs is constructed from negacyclic codes of length $n=\frac{q^2+1}{a}$, where $q$ is an odd prime power, $a=\frac{m^2+1}{2}$ and $m$ is an odd integer. Some new entanglement-assisted quantum maximum distance separable (EAQMDS) codes are obtained in the sense that their parameters are not covered by the previously known ones.