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A138367 Count of post-period decimal digits up to which the rounded n-th convergent to sqrt(5) agrees with the exact value.

%I A138367 #12 Mar 30 2020 06:16:17
%S A138367 0,2,4,5,6,7,8,10,8,12,14,14,16,18,19,20,21,23,24,24,26,28,29,30,31,
%T A138367 33,33,34,35,37,39,40,41,42,44,44,46,47,48,49,51,53,53,55,56,57,59,60,
%U A138367 60,61,64,65,66,68,69,70,72,73,74,75,76,77,79,80,81,83,83,85,85,88,89,90,91,92
%N A138367 Count of post-period decimal digits up to which the rounded n-th convergent to sqrt(5) agrees with the exact value.
%C A138367 This is a measure of the quality of the n-th convergent to A002163 if the convergent and the exact value are compared rounded to an increasing number of digits.
%C A138367 The sequence of rounded values of sqrt(5) is 2, 2.2, 2.24, 2.236, 2.2361, 2.23607, 2.236068, 2.2360680 etc, and the n-th convergent (provided by A001077 and A001076) is to be represented by its equivalent sequence.
%C A138367 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 A138367 For n=3, the 3rd convergent is 161/72 = 2.236111111..., with a sequence of rounded representations 2, 2.2, 2.24, 2.236, 2.2361, 2.23611, 2.236111, 2.2361111 etc.
%e A138367 Rounded to 1, 2, 3, or 4 post-period decimal digits, this is the same as the rounded version of the exact sqrt(5), but disagrees if both are rounded to 5 decimal digits, where 2.23607 <> 2.23611.
%e A138367 So a(3) = 4 (digits), the maximum rounding level of agreement.
%Y A138367 Cf. A138335, A138336, A138337, A138339, A138343, A138366, A138369, A138370.
%K A138367 nonn,base
%O A138367 1,2
%A A138367 _Artur Jasinski_, Mar 17 2008
%E A138367 Definition and values replaced as defined via continued fractions by _R. J. Mathar_, Oct 01 2009