recursion – simple recursive function

F[2 x_] : = f[x]
F[1] : = 3
F[0] : = 0
F[2 x_ + 1] : = f[x] + f[x + 1]
a[x_] : = f[x]/ F[x + 1]

Will it work as a recursive function?
I think there's something wrong with that because every integer will have an output of 3

any help would be appreciated, thank you very much