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We find an unexpected scaling in the correlation energy of artificial atoms, i.e., harmonically confined two-dimensional quantum dots. The scaling relation is found through extensive numerical examinations including Hartree-Fock, variational quantum Monte Carlo, density-functional, and full configuration-interaction calculations. We show that the correlation energy, i.e., the true ground-state total energy subtracted by the Hartree-Fock total energy, follows a simple function of the Coulomb energy, confimenent strength and, the number of electrons. We find an analytic expression for this function, as well as for the correlation energy per particle and for the ratio between the correlation and total energies. Our tests for independent diffusion Monte Carlo and coupled-cluster results for quantum dots -- including open-shell data -- confirm the generality of the obtained scaling. As the scaling is also well applicable to $\gtrsim$ 100 electrons, our results give interesting prospects for the development of correlation functionals within density-functional theory., Comment: Accepted to Journal of Physics: Condensed Matter