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A138343 Count of post-period decimal digits up to which the rounded n-th convergent to Pi agrees with the exact value.

%I A138343 #12 Mar 30 2020 08:43:33
%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,
%T A138343 27,29,30,29,31,33,34,37,39,39,40,42,43,44,45,45,47,46,49,49,51,52,52,
%U A138343 54,55,56,55,56,57,59,58,59,60,61,61,63,64,64,65,65,66,67,67,68,69,70,71,72,72
%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 convergent and the exact value are compared rounded to an increasing number of digits. (This is similar to A084407 which compares the truncated/floored values).
%C A138343 The sequence of rounded values of Pi is 3, 3.1, 3.14, 3.142, 3.1416, 3.14159, 3.141593, 3.1415927 etc, and the n-th convergent (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 if compared at the same level of rounding. Counting only post-period digits (which is 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 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 of the exact Pi, but disagrees if both are rounded to 7 decimal digits, where 3.1415927 <> 3.1415929.
%e A138343 So a(3) = 6 (digits), the maximum rounding level of agreement.
%Y A138343 Cf. A138335, A138336, A138337, A138339.
%K A138343 nonn,base
%O A138343 0,2
%A A138343 _Artur Jasinski_, Mar 16 2008
%E A138343 Definition and values replaced as defined via continued fractions by _R. J. Mathar_, Oct 01 2009