%I A138343
%S A138343 0,2,3,6,8,9,8,10,10,11,11,13,15,15,16,15,17,17,18,19,20,23,24,23,26,27,
%T A138343 29,30,29,31,33,34,37,39,39,40,42,43,44,45,45,47,46,49,49,51,52,52,54,
               55,
%U A138343 56,55,56,57,59,58,59,60,61,61,63,64,64,65,65,66,67,67,68,69,70,71,72,
               72
%N A138343 Count of post-period decimal digits up to which the rounded n-th convergent 
               to Pi agrees with the exact value.
%C A138343 This is a measure of the quality of the n-th convergent to A000796 if 
               the
%C A138343 convergent and the exact value are compared rounded
%C A138343 to an increasing number of digits. (This is similar to A084407 which 
               compares
%C A138343 the truncated/floored values). The sequence of rounded values of Pi is
%C A138343 3, 3.1, 3.14, 3.142, 3.1416, 3.14159, 3.141593, 3.1415927 etc, and the 
               n-th convergent
%C A138343 (provided by A002485 and A002486) is to be represented by its equivalent 
               sequence.
%C A138343 a(n) represents the maximum number of post-period digits of the two sequences
%C A138343 if compared at the same level of rounding. Counting only post-period 
               digits (which is
%C A138343 one less than the full number of decimal digits) is just a convention 
               taken from A084407.
%e A138343 For n=3, the 3rd convergent is 355/113 = 3.141592920353.., with a sequence 
               of rounded
%e A138343 representations 3, 3.1, 3.14, 3.142, 3.1416, 3.141593, 3.1415929, 3.14159292 
               etc.
%e A138343 Rounded to 1, 2, 3, 4, 5 or 6 post-period decimal digits, this is the 
               same as the rounded version
%e A138343 of the exact Pi, but disagrees if both are rounded to 7 decimal digits, 
               where 3.1415927 <> 3.1415929.
%e A138343 So a(n=3)= 6 (digits), the maximum rounding level of agreement.
%Y A138343 Cf. A138335, A138336, A138337, A138338, A138339.
%Y A138343 Sequence in context: A140266 A140265 A127293 this_sequence A139371 A079338 
               A047405
%Y A138343 Adjacent sequences: A138340 A138341 A138342 this_sequence A138344 A138345 
               A138346
%K A138343 nonn,base
%O A138343 0,2
%A A138343 Artur Jasinski (grafix(AT)csl.pl), Mar 16 2008
%E A138343 Definition and values replaced as defined via continued fractions - R. 
               J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 01 2009