A conjectured good rational approximations to the cubic irrationality of the real root of,

                                2     2                  3
    64 + 144 x (2 c + 1) + 108 x  (3 c  + 3 c + 1) + 27 x  (2 c + 1) = 0, 

for  positive intgers c, with amazing irrationality measures as c gets bigger




                              By Shalosh B. Ekhad



         Consider the real root of the cubic equation (with a positive)



                                  2     2                  3
      64 + 144 x (2 c + 1) + 108 x  (3 c  + 3 c + 1) + 27 x  (2 c + 1) = 0



                        that according to Cardano equals



   1/3          5        4        3               2 1/2       2           2 1/2
2 4    (c (-54 c  - 162 c  - 180 c  + 2 c ((c + 1) )    - 90 c  + ((c + 1) )

                 1/3                         3       2              2/3   /
     - 19 c - 1))   /(3 (2 c + 1)) + 2 c (9 c  + 18 c  + 11 c + 2) 4     /  (3
                                                                        /

                       5        4        3               2 1/2       2
    (2 c + 1) (c (-54 c  - 162 c  - 180 c  + 2 c ((c + 1) )    - 90 c

                                              2
               2 1/2             1/3    4 (3 c  + 3 c + 1)
     + ((c + 1) )    - 19 c - 1))   ) - ------------------
                                           3 (2 c + 1)



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



                          /(c x + 1) (1 + (c + 1) x)\n
                          |-------------------------|
                          \            x            /
                          ----------------------------
                                       2/3
                                      x



                               let's call it X(n)



                           that in Maple notation is



((c*x+1)*(1+(c+1)*x)/x)^n/x^(2/3)


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





X(n) = 9*n*(2*n-1)*(2*c+1)/(3*n-1)/(3*n+1)*X(n-1)-9*n*(n-1)/(3*n-1)/(3*n+1)*X(n
-2)


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



                          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\
    -mentioned cubic irrationality, namely the real root of



                                  2     2                  3
      64 + 144 x (2 c + 1) + 108 x  (3 c  + 3 c + 1) + 27 x  (2 c + 1) = 0



Not only that, these seem to give effective irrationality measures, that sta\
    rting with c=4 are better than Liouville's irratioanlity measure of 3



            The estimated irrationality measure for the case c=4 is



2.91671189692388132500469650131481711519220015086338875796898785652487333243\
    7465334476698221121493387



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



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



2.62142830520850111722825132771928190392903298875791069032549231209765301155\
    6238400702347077250011859

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



2.35373664843180663073757986375881023985617960073296259684672903697139392716\
    1655887277018248693710482

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



2.24435656427019258831196836391360623093579999326384378403454426981382307673\
    2081522524069181245322821

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



2.18624329935829019141483121953750310542157530895252891118020698341682766580\
    0235444966032459245702170

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



2.15044659611568879165606626307703764750110409260728203791726569488432883345\
    1297254010213171141808498



                    as you can see, it seems to go down to 2