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Sandwich structures with honeycomb core are widely used in the lightweight design and impact energy absorption applications in automotive, sporting, and aerospace industries. Recently, the auxetic honeycombs with negative Poisson's ratio attract a substantial attention for different engineering products. In this study, we implement Additive Manufacturing technology, experimental testing, and Finite Element Analysis (FEA) to design and investigate the mechanical behavior of a novel unit cell for sandwich structure core. The new core model contains the conventional and auxetic honeycomb cells beside each other to create a Hybrid Honeycomb (HHC) for the sandwich structure. The different designs of unit cells with the same volume fraction of 15% are 3D-printed using Fused Deposition Modeling technique, and the comparative study on the mechanical behavior of conventional honeycomb, auxetic honeycomb, and HHC structures is conducted. The quasi-static uniaxial compression tests are performed on the printed samples to investigate the mechanical behavior of the printed structures. The deformation and failure modes of the different designs are studied at the cell level utilizing FEA of the compression test and experimental observation. The compressive strength of the different design is measured using three experimental tests. The new HHC unit cell design shows significantly higher mechanical properties than the auxetic and the conventional designs. Modifying the design variables of hybrid cellular core structure allows us to tailor the mechanical properties and deformation pattern in macro level to achieve the desired mechanical properties in sandwich structures.