OFFSET
0,1
COMMENTS
The decimal expansion of the fraction 9099999999923 / 27500000000000 is 0.3309090909062909090... (with the "90" digit pairs infinitely repeating).
One way to write this fraction as a simple continued fraction with single-digit terms is
[ 0; 3, 3, 0, 9, 0, 9, 0, 9, 0, 9, 0, 6, 2,
9, 0, 9, 0, 9, 0, ..., 9, 0, 9, 0, 9, 0,
6, 1, 2, 6, 1, 1, 1, 9, 0, 9, 0, 5, 2, 1, 4 ]
where the second line contains the repeating term pair "9, 0," a total of 524727 times. The digits of the decimal expansion perfectly agree with the first two lines of the terms of the continued fraction. So we have 1049483 terms in the continued fraction with 1049468 digits of agreement.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
MATHEMATICA
(* Mma code from Michael Trott to check this: *)
cf = {0, 3, 3, 0, 9, 0, 9, 0, 9, 0, 9, 0, 6, 2,
Sequence @@ Flatten[Table[{9, 0}, {524727}]],
6, 1, 2, 6, 1, 1, 1, 9, 0, 9, 0, 5, 2, 1, 4};
In[271]:= fcf = FromContinuedFraction[cf]
In[309]:= rds = RealDigits[fcf, 10, 1049468, 0][[1]];
In[311]:= Take[cf, 1049468] === rds
Out[311]= True (* end of program *)
RealDigits[9099999999923 / 27500000000000, 10, 110][[1]] (* Vincenzo Librandi, Sep 03 2013 *)
CROSSREFS
KEYWORD
AUTHOR
Jon E. Schoenfield, Apr 09 2010
EXTENSIONS
Edited by Charles R Greathouse IV, Aug 02 2010
STATUS
approved