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It is a well-known fact that a cluster of nucleons can be formed in the interior of an atomic nucleus, and such clusters may occupy molecular-like orbitals, showing characteristics similar to normal molecules consisting of atoms. Chemical molecules having a linear alignment are commonly seen in nature, such as carbon dioxide. A similar linear alignment of the nuclear clusters, referred to as linear-chain cluster state (LCCS), has been studied since the 1950s, however, up to now there is no clear experimental evidence demonstrating the existence of such a state. Recently, it was proposed that an excess of neutrons may offer just such a stabilizing mechanism, revitalizing interest in the nuclear LCCS, specifically with predictions for their emergence in neutron-rich carbon isotopes. Here we present the experimental observation of α-cluster states in the radioactive 14C nucleus. Using the Be10+α resonant scattering method with a radioactive beam, we observed a series of levels which completely agree with theoretically predicted levels having an explicit linear-chain cluster configuration. We regard this as the first strong indication of the linear-chain clustered nucleus.