For unknowns A,B,C that query seems out of reach of WolframAlpha.
WolframAlpha is able to provide some results for queries like this
RSolve[{a[n]==2*a[n-1]^3/(4+a[n-1]),a[0]==1/2},a[n],n]
That might help you get started, but that result may not be in the form that you desire