  We give new rational blowdown constructions of exotic CP^2#n(-CP^2) (5\leq
n\leq 9) without using elliptic fibrations. We also show that our 4-manifolds
admit handle decompositions without 1- and 3-handles, for 7\leq n\leq 9. A
strategy for rational blowdown constructions of exotic CP^2#n(-CP^2) (1\leq
n\leq 4) is also proposed.
