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A138366 Count of post-period decimal digits up to which the rounded n-th convergent to exp(1) agrees with the exact value.

%I A138366 #12 Mar 30 2020 06:17:18
%S A138366 0,1,0,1,2,3,4,5,5,6,7,7,8,9,10,12,12,13,14,16,15,16,19,18,20,22,22,
%T A138366 24,25,25,26,27,28,30,32,32,32,35,36,36,39,39,41,43,43,44,46,46,48,50,
%U A138366 50,52,52,54,56,57,58,59,61,61,63,65,64,67,69,69,71,72,73,74,77,77,79,80,81,83
%N A138366 Count of post-period decimal digits up to which the rounded n-th convergent to exp(1) agrees with the exact value.
%C A138366 This is a measure of the quality of the n-th convergent to E = A001113 if the convergent and the exact value are compared rounded to an increasing number of digits.
%C A138366 The sequence of rounded values of exp(1) is 3, 2.7, 2.72, 2.718, 2.7183, 2.71828, 2.718282, 2.7182818 etc, and the n-th convergent (provided by A007676 and A007677) is to be represented by its equivalent sequence.
%C A138366 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 A138366 For n=6, the 6th convergent is 106/39 = 2.7179487.., with a sequence of rounded representations 3, 2.7, 2.72, 2.718, 2.7179, 2.71795, 2.717949, etc.
%e A138366 Rounded to 1, 2, or 3 post-period decimal digits, this is the same as the rounded version of the exact E, but disagrees if both are rounded to 4 decimal digits, where 2.7183 <> 2.7179.
%e A138366 So a(6) = 3 (digits), the maximum rounding level of agreement.
%Y A138366 Cf. A138335, A138336, A138337, A138339, A138343, A138367, A138369, A138370.
%K A138366 nonn,base
%O A138366 1,5
%A A138366 _Artur Jasinski_, Mar 17 2008
%E A138366 Definition and values replaced as defined via continued fractions by _R. J. Mathar_, Oct 01 2009