Find the generating function for the sequence ${1, 2, 1, 4, 1, 8, ...}$

Question

I have tried a few basic technique. But it seems still can't come up with a closed form for this sequence ${1, 2, 1, 4, 1, 8, ...}$. See if anyone can give me a hint?

Answer 1

Hint:

• Find the generating function for 1,1,1,1, ...
• Deduce the generating function for 1,0,1,0,1,0,1, ...
• Find the generating function for 2,4,8,16, ...
• Deduce the generating function for 2,0,4,0,8,0,16, ...
• Deduce the generating function for 0,2,0,4,0,8,0,16, ...
• Deduce the generating function you are interested in.