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