Let f(n) denotes the number of different ways the positive integer n can be expressed as the sum of 1's and 2's. For example f(3)=3, since 3=1+1+1=1+2=2+1 f(4) = 5, since 4=1+1+1+1=1+1+2=1+2+1=2+1+1=2+2 Note that order of the appearance of 1's and 2's is important. f(f(6)) is equal to (A) 377 (C) 321 (B) 302 (D) 350
My try: Actually, I was able to solve the problem myself by manually calculating f(6) then f(13) using some basic concepts of permutations.
However, I wondered if we could think of some type of general formula (kind of functional mapping) or recurrence for f(n).
My try at my thought: I thought of making cases for n even and n odd and tried making a recurrence for f(n-1) but failed.
Thus, I need some help so as to how to proceed about what I am thinking.
