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We present what we believe to be the first known example of an exactquasiperiodic localized stable solution with spatially symmetriclarge-amplitude oscillations in a non-integrable Hamiltonian lattice model. Themodel is a one-dimensional discrete nonlinear Schr\"odinger equation withalternating on-site energies, modelling e.g. an array of optical waveguideswith alternating widths. The solution bifurcates from a stationary discrete gapsoliton, and in a regime of large oscillations its intensity oscillatesperiodically between having one peak at the central site, and two symmetricpeaks at the neighboring sites with a dip in the middle. Such solutions, termed'pulsons', are found to exist in continuous families ranging arbitrarily closeboth to the anticontinuous and continuous limits. Furthermore, it is shown thatthey may be linearly stable also in a regime of large oscillations.Comment: 4 pages, 4 figures, to be published in Phys. Rev. E. Revised version:  change of title, added Figs. 1(b),(c), 4 new references + minor  clarification

Quasiperiodic localized oscillating solutions in the discrete nonlinear Schrödinger equation with alternating on-site potential