Recently,　the　echo　state　networks　(ESNs)　have　been　widely　used　for　time　series　prediction.　To　meet　the　demand　of　actual　applications　and　avoid　the　overfitting　issue,　the　online　sequential　ESN　with　sparse　recursive　least　squares　(OSESN-SRLS)　algorithm　is　proposed.　Firstly,　the　l(0)　and　l(1)norm　sparsity　penalty　constraints　of　output　weights　are　separately　employed　to　control　the　network　size.　Secondly,　the　sparse　recursive　least　squares　(SRLS)　algorithm　and　the　subgradients　technique　are　combined　to　estimate　the　output　weight　matrix.　Thirdly,　an　adaptive　selection　mechanism　for　the　l(0)　or　l(1)　norm　regularization　parameter　is　designed.　With　the　selected　regularization　parameter,　it　is　proved　that　the　developed　SRLS　shows　comparable　or　better　performance　than　the　regular　RLS.　Furthermore,　the　convergence　of　OSESN-SRLS　is　theoretically　analyzed　to　guarantee　its　effectiveness.　Simulation　results　illustrate　that　the　proposed　OSESN-SRLS　always　outperforms　other　existing　ESNs　in　terms　of　estimation　accuracy　and　network　compactness.　(C)　2019　Elsevier　Ltd.　All　rights　reserved.