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Since the existing inverse-matrix-free extreme learning machine (IF-ELM) only works well in batched way, this paper extends it into its inverse-matrix-free online sequential version called the inverse-matrix-free online sequential extreme learning machine (IOS-ELM). When the proposed algorithm increases the training samples, the Sherman Morrison Woodbury formula is used to update the model, and the newly added hidden layer output weights are directly calculated to avoid the iterative calculation of output weight of analysed training samples. The detailed derivations of the proposed machine IOS-ELM are accordingly given. The experimental results on different types and sizes of datasets show that IOS-ELM indeed is very suitable for the datasets which are gradually generated in an online way, in the sense of both fast training and promising performance.