Case A

In this situation, a recurrence is already available in the system. It is readily solved.

> eq[0]:=collect(eval(subs(f=unapply(f(n),n,u),op(remove(has,sys[2],diff)))),f,factor);

> normal(LREtools[hypergeomsols](eq[0],f(n),{},output=basis),expanded);

> solve(expand({subs(n=1,%)=2*u,subs(n=2,%)=2*u,subs(n=3,%)=2*u}),{_C[1],_C[2]});

> rsol:=normal(subs(%,%%),expanded);