Vision Transformers (ViTs) overpass Convolutional Neural Networks in processing incomplete inputs because they do not require the imputation of missing values. Therefore, ViTs are well suited for sequential decision-making, e.g. in the Active Visual Exploration problem. However, they are computationally inefficient because they perform a full forward pass each time a piece of new sequential information arrives. To reduce this computational inefficiency, we introduce the TOken REcycling (TORE) modification for the ViT inference, which can be used with any architecture. TORE divides ViT into two parts, iterator and aggregator. An iterator processes sequential information separately into midway tokens, which are cached. The aggregator processes midway tokens jointly to obtain the prediction. This way, we can reuse the results of computations made by iterator. Except for efficient sequential inference, we propose a complementary training policy, which significantly reduces the computational burden associated with sequential decision-making while achieving state-of-the-art accuracy.