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Ti3CrCu4 is a new ternary compound which crystallizes in the tetragonal Ti3Pd5 structure type. The Cr atoms form square nets in the a-b plane (a = 3.124 Å) which are separated by an unusually large distance c = 11.228 Å along the tetragonal axis, thus forming a -2-D Cr-sublattice. The paramagnetic susceptibility is characterized by a low effective moment, μeff = 1.1 μB, a low paramagnetic Curie temperature θP (below 7 K) and a temperature independent χ0 = 6.7 x 10−4 emu/mol. The magnetization at 1.8 K increases rapidly with field nearly saturating to 0.2 μB/f.u. The zero field heat capacity C/T shows an upturn below 7 K (∼190 mJ/mol K2 at ∼0.1K) which is suppressed in applied magnetic fields and interpreted as suggesting the presence of spin fluctuations. The resistivity at low temperatures shows non-Fermi liquid behavior. Overall, the experimental data thus reveal an unusual magnetic state in Ti3CrCu4, which likely has its origin in the layered nature of the Cr sub-lattice and ferromagnetic spin fluctuations. Density functional theoretical calculations reveal a sharp Cr density of states peak just above the Fermi level, indicating the propensity of Ti3CrCu4 to become magnetic.