O10-131
> restart;
> u:=n->n+a*sqrt(n^2+n+1)+b*sqrt(n^2-n+2)+c*sqrt(n^2+n+3);

> d:=series(u(n),n=infinity,3);

                                             11 c   7 b   3 a
                                             ---- + --- + ---
                                              8      8     8
  d := (1 + a + b + c) n + c/2 + a/2 - b/2 + ----------------
                                                    n

           7 b   11 c   3 a
           --- - ---- - ---
           16     16    16       1
         + ---------------- + O(----)
                   2              3
                  n              n

> p:=convert(d,polynom);

                                             11 c   7 b   3 a
                                             ---- + --- + ---
                                              8      8     8
  p := (1 + a + b + c) n + c/2 + a/2 - b/2 + ----------------
                                                    n

           7 b   11 c   3 a
           --- - ---- - ---
           16     16    16
         + ----------------
                   2
                  n

> sys:={seq(coeff(p,n,k),k=-1..1)};

                                              11 c   7 b   3 a
      sys := {1 + a + b + c, c/2 + a/2 - b/2, ---- + --- + ---}
                                               8      8     8

> s:=solve(sys,{a,b,c});

                  s := {c = 5/8, b = -1/2, a = -9/8}

> subs(s,d);

                              7        1
                          - ----- + O(----)
                                2       3
                            16 n       n