If f(x) = n=0,anxn what is the generating function for the sequencer, a0 + a0, a1 +

Question:

If f(x) = ∑∞n=0,anxn what is the generating function for the sequencer, a0 + a0, a1 + a2, a2 + a3, . . . ? What is the generating function for the sequence a0, a0 + a1, a2 + a2 + a1, a2 + a3 + a3, a2 + a3 + a4, . .. ? What is the generating function for the sequence a0/4 + a0/2, a1/4 + a1/2 + a2/4, a1/4 + a2/2 + a3/4, ... ?