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[en] Standing sausage modes in flare loops are important for interpreting quasi-periodic pulsations (QPPs) in solar flare light curves. We propose an inversion scheme that consistently uses their periods P and damping times τ to diagnose flare loop parameters. We derive a generic dispersion relation governing linear sausage waves in pressure-less straight tubes, for which the transverse density inhomogeneity takes place in a layer of arbitrary width l and is of arbitrary form. We find that P and τ depend on the combination of where R is the loop radius, L is the looplength, vAi is the internal Alfvén speed, and ρi/ρe is the density contrast. For all the density profiles examined, P and τ experience saturation when L/R ≫ 1, yielding an inversion curve in the space with a specific density profile when L/R is sufficiently large. When applied to a spatially unresolved QPP event, the scheme yields that R/vAi is the best constrained, whereas l/R corresponds to the other extreme. For spatially resolved QPPs, while L/R ≫ 1 cannot be assumed beforehand, an inversion curve remains possible due to additional geometrical constraints. When a spatially resolved QPP event involves another mode, as is the case for a recent event, the full set of can be inferred. We conclude that the proposed scheme provides a useful tool for magneto-seismologically exploiting QPPs.