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We propose a new paradigm for encoding the geometry of point cloud sequences, where the convolutional neural network (CNN) which estimates the encoding distributions is optimized on several frames of the sequence to be compressed. We adopt lightweight CNN structures, we perform training as part of the encoding process, and the CNN parameters are transmitted as part of the bitstream. The newly proposed encoding scheme operates on the octree representation for each point cloud, encoding consecutively each octree resolution layer. At every octree resolution layer, the voxel grid is traversed section-by-section (each section being perpendicular to a selected coordinate axis) and in each section the occupancies of groups of two-by-two voxels are encoded at once, in a single arithmetic coding operation. A context for the conditional encoding distribution is defined for each two-by-two group of voxels, based on the information available about the occupancy of neighbor voxels in the current and lower resolution layers of the octree. The CNN estimates the probability distributions of occupancy patterns of all voxel groups from one section in four phases. In each new phase the contexts are updated with the occupancies encoded in the previous phase, and each phase estimates the probabilities in parallel, providing a reasonable trade-off between the parallelism of processing and the informativeness of the contexts. The CNN training time is comparable to the time spent in the remaining encoding steps, leading to competitive overall encoding times. Bitrates and encoding-decoding times compare favorably with those of recently published compression schemes.