O15-064
> restart:
> u:=n->(1/(6*n+1)+3/2/(6*n+2)+1/4/(6*n+3)-1/8/(6*n+5))/(64^n);

                  1             1             1             1
               ------- + 3/2 ------- + 1/4 ------- - 1/8 -------
               6 n + 1       6 n + 2       6 n + 3       6 n + 5
     u := n -> -------------------------------------------------
                                        n
                                      64

> evalf(sum(u(n),n=0..100),4);

                                1.814

> #u(n)<1/2^(6*n) donc R_n est de l'ordre de 2^(-6(n+1))
> i1:=int(1/(1-x+x^2),x=0..1/2);i2:=int(x/(1-x+x^2),x=0..1/2);i3:=int(x/(1-x^2),x=0..1/2);

                         i1 := 1/9 Pi sqrt(3)


              i2 := 1/2 ln(3) - ln(2) + 1/18 Pi sqrt(3)


                       i3 := ln(2) - 1/2 ln(3)

> zero:=expand(Pi/a/sqrt(b)-c*i1-d*i2-e*i3);

             Pi
  zero := --------- - 1/9 c Pi sqrt(3) - 1/2 d ln(3) + d ln(2)
          a sqrt(b)

         - 1/18 d Pi sqrt(3) - e ln(2) + 1/2 e ln(3)

> zero1:=expand(Pi/a/sqrt(b)-c*i1-d*i2-d*i3);#e=d pour faire partir les ln

                  Pi
      zero1 := --------- - 1/9 c Pi sqrt(3) - 1/18 d Pi sqrt(3)
               a sqrt(b)

> # a=2;b=3;c=d=e=1
> p:=expand(simplify((1-x^6)*(1/(1-x+x^2)+x/(1-x+x^2)+x/(1-x^2))));

                             2      4
                       p := x  - 2 x  + 3 x + 1

> # On intègre TAT la SE de TG : -2*x^(6*n+4)+x^(6*n+2)+3*x^(6*n+1)+x^(6*n) ce qui donne sigma(u(n)/2)
> int(p/(1-x^6),x=0..1/2);

                            1/6 Pi sqrt(3)

> 2*evalf(%);# OK

                             1.813799365

> # v_n=u_(2*n)+u_(2*n+1)