Sequential pattern mining is an important data mining task with applications in basket analysis, world wide web, medicine and telecommunication. This task is challenging because sequence databases are usually large with many and long sequences and the number of possible sequential patterns to mine can be exponential. We proposed a new sequential pattern mining algorithm called FAST which employs a representation of the dataset with indexed sparse id-lists to fast counting the support of sequential patterns. We also use a lexicographic tree to improve the efficiency of candidates generation. FAST mines the complete set of patterns by greatly reducing the effort for support counting and candidate sequences generation. Experimental results on artificial and real data show that our method outperforms existing methods in literature up to an order of magnitude or two for large datasets.