Due to the wide range of time scales involved in the ordinary differential equations (ODEs) describing chemical reaction kinetics, multidimensional numerical simulation of chemical reactive flows using detailed combustion mechanisms is computationally expensive. To confront this issue, this article presents an economic data-driven tabulation algorithm for fast combustion chemistry integration. It uses the recurrent neural networks (RNNs) to construct the tabulation from a series of current and past states to the next state, which takes full advantage of RNN in handling long-term dependencies of time series data. The training data are first generated from direct numerical integrations to form an initial state space, which is divided into several subregions by the K-means algorithm. The centroid of each cluster is also determined at the same time. Next, an Elman RNN is constructed in each of these subregions to approximate the expensive direct integration, in which the integration routine obtained from the centroid is regarded as the basis for a storing and retrieving solution to ODEs. Finally, the alpha-shape metrics with principal component analysis (PCA) are used to generate a set of reduced-order geometric constraints that characterize the applicable range of these RNN approximations. For online implementation, geometric constraints are frequently verified to determine which RNN network to be used to approximate the integration routine. The advantage of the proposed algorithm is to use a set of RNNs to replace the expensive direct integration, which allows to reduce both the memory consumption and computational cost. Numerical simulations of a H2/CO-air combustion process are performed to demonstrate the effectiveness of the proposed algorithm compared to the existing ODE solver.