2
The Apery Constant Inspired by the coefficient of , t ,

     in the Zudilin-Straub transform of,

      n
    -----
     \                  2
      )   binomial(n, k)  binomial(n + k, k)
     /
    -----
    k = 0



                              By Shalosh B. Ekhad



      Let RF(t,k), as usual, be the raising factorial, t*(t+1)...(t+k-1)



We are interested in the limit, as n goes to infinity, of the coefficient of ,

     2
    t ,  in the following quantity



                                            3           2
                       RF(1 + t, n + k) (k!)  ((n - k)!)
                    ----------------------------------------
                                3                 2
                    RF(1 + t, k)  RF(1 - t, n - k)  (n + k)!



              / 1/2      \
              |5         |
and k equals, |---- - 1/2| n, that in floating-point is,
              \ 2        /

    0.61803398874989484820 n



Let's call this constant c (it can also be described directly in terms of ha\
    rmonic-type expressions). Using this definition the convergence is extre\
    mely slow.



    We will show how to compute it much faster, with exponential error-rate



     Let A(n) be the sequence of integers satisfying the linear recurrence



             2              2                                2
     -(n + 1)  A(n) + (-11 n  - 33 n - 25) A(n + 1) + (n + 2)  A(n + 2) = 0



                             and in Maple notation



-(n+1)^2*A(n)+(-11*n^2-33*n-25)*A(n+1)+(n+2)^2*A(n+2) = 0


                       Subject to the initial condition



A(1) = 3, A(2) = 19




    Let  B(n) be the sequence satisfying the INHOMOGENEOUS recurrence i.e.



-(n+1)^2*B(n)+(-11*n^2-33*n-25)*B(n+1)+(n+2)^2*B(n+2) = C(n)


                          with the initial conditions



B(1) = 9, B(2) = 245/4


             where the right side, C(n), satisfies the recurrence





-2*(29*n+69)*(2*n+3)/(n+3)/(29*n+40)*C(n)+C(n+1) = 0




                          with the initial conditions



C(1) = 69


   Note that it is very fast to compute many terms of C(n) and hence of B(n)



             The ratios B(n)/A(n) tend very fast to the constant c



            Here it is to 30 digits, 3.28986813369645287294483033329



                                                    2
                                                  Pi
                           This is most probably, ---
                                                   3



To illustrate how fast the convergence is, and also as check, let's compute \
    the n=5000 and n=10000 values of the above sequence, whose limit is our \
    C



                             3.28892137, 3.28939477

     as you can see the convergence is extremely slow using the definition



            -------------------------------------------------------



        This ends this paper that took, 67126.422, seconds to generate.