2      1/2
A conjectured good rational approximations to , -3 c - 3/2 - 3 c  - 3 c   ,

    for positive intgers c,as good as from the continued fraction



                              By Shalosh B. Ekhad



                                                        2     1/2
               Regarding the constant, -3 c - 3/2 - 3 (c  + c)





Consider the recurrence satisfied by the contour-integral, around x=0, of th\
    e function



                          /(c x + 1) (1 + (c + 1) x)\n
                          |-------------------------|
                          \            x            /
                          ----------------------------
                                       1/2
                                      x



                               let's call it X(n)



                           that in Maple notation is



((c*x+1)*(1+(c+1)*x)/x)^n/x^(1/2)


According to the famous Almkvist-Zeilberger algorithm, it satisfies the recu\
    rrence





X(n) = 4*(2*c+1)*n/(2*n+1)*X(n-1)-4*n*(n-1)/(2*n-1)/(2*n+1)*X(n-2)


       Let a(n), b(n), be the sequence that satisfy that recurrence with



                          with the initial conditions



                               a(0) = 0, a(1) = 1



                               b(0) = 1, b(1) = 0



We conjecture that the limit of a(n)/b(n) as n goes to infinity is the above\

                                 2     1/2
    -mentioned, -3 c - 3/2 - 3 (c  + c)



Not only that, these seem to give irrationality measure 2, just like the one\
     coming from the continued fraction.



            The estimated irrationality measure for the case c=4 is



1.99915073193908718698254299933544062813603232407152470164117399101580021456\
    0315642671109465789245188



   Let's make c bigger and see how the effective irrationaliy measure shrinks



            For c=, 11, the irrationality measure is estimated to be



1.99947328572055162849893869342398793095390713732342467397798302362273506450\
    1946315449219698264109302

           For c=, 101, the irrationality measure is estimated to be



1.99930513358741971399617960099126194140551487959795751388712743331141408519\
    0520764095345856703052627

           For c=, 1001, the irrationality measure is estimated to be



1.99922365115818617422396056985989868210486621832151800504192411476209369450\
    8870902283979428907551897

          For c=, 10001, the irrationality measure is estimated to be



1.99917722762411759657889501182835940437093644308950175600429207795684051522\
    4100808748538153338085936

          For c=, 100001, the irrationality measure is estimated to be



1.99914735812240415317130504063759387171912530025823994863811840697064988221\
    9235368483911054561507306



                    as you can see, it seems to be 2