G-frames　generalize　frames　in　Hilbert　spaces.　The　literatures　show　that　g-frames　and　frames　share　many　similar　properties,　while　they　behave　differently　in　redundancy　and　perturbation　properties.　Interestingly,　g-frames　have　been　extensively　studied,　but　g-frame　sequences　have　not.　This　problem　is　nontrivial　since　a　g-frame　and　a　frame　both　involve　all　vectors　in　the　same　Hilbert　space,　while　a　g-frame　sequence　and　a　frame　sequence　do　not.　They　involve　different　linear　spans.　Using　the　synthesis　and　Gram　matrix　methods,　we　in　this　paper　characterize　g-frame　sequences　and　g-Riesz　sequences;　obtain　the　Pythagorean　theorem　for　g-orthonormal　systems.　These　results　recover　several　known　results　and　lead　to　some　new　results　on　g-frames.