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A novel dose calculation approach was designed based on the application of LSTM network that processes the 3D patient/phantom geometry as a sequence of 2D computed tomography input slices yielding a corresponding sequence of 2D slices that forms the respective 3D dose distribution. LSTM networks can propagate information effectively in one direction, resulting in a model that can properly imitate the mechanisms of proton interaction in matter. The study is centered on predicting dose on a single pencil beam level, avoiding the averaging effects in treatment plans comprised of thousands pencil beams. Moreover, such approach allows straightforward integration into today's treatment planning systems' inverse planning optimization process. The ground truth training data was prepared with Monte Carlo simulations for both phantom and patient studies by simulating different pencil beams impinging from random gantry angles through the patient geometry. For model training, 10'000 Monte Carlo simulations were prepared for the phantom study, and 4'000 simulations were prepared for the patient study. The trained LSTM model was able to achieve a 99.29 % gamma-index pass rate ([0.5 %, 1 mm]) accuracy on the set-aside test set for the phantom study, and a 99.33 % gamma-index pass rate ([0.5 %, 2 mm]) for the set-aside test set for the patient study. These results were achieved for each pencil beam in 6-23 ms. The average Monte Carlo simulation run-time using Topas was 1160 s. The generalization of the model was verified by testing for 5 previously unseen lung cancer patients. LSTM networks are well suited for proton therapy dose calculation tasks. However, further work needs to be performed to generalize the proposed approach to clinical applications, primarily to be implemented for various energies, patient sites, and CT resolutions/scanners.