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A138371 Count of post-period decimal digits up to which the rounded n-th convergent to A058265 agrees with the exact value.

%I A138371 #18 Mar 29 2020 17:13:13
%S A138371 0,1,2,5,8,7,10,11,10,12,15,17,17,17,20,21,22,23,25,26,28,30,29,30,31,
%T A138371 31,32,32,34,35,35,36,36,38,40,40,42,42,42,43,44,43,45,46,47,47,49,52,
%U A138371 51,52,54,54,55,57,59,59,60,60,60,61,61,62,62,64,64,66,67,69,71,73,74
%N A138371 Count of post-period decimal digits up to which the rounded n-th convergent to A058265 agrees with the exact value.
%C A138371 This is a measure of the quality of the n-th convergent to the tribonacci constant A058265 if the convergent and the exact value are compared rounded to an increasing number of digits. The sequence of rounded values of A058265 is 2, 1.8, 1.84, 1.839, 1.8393, 1.83929, 1.839287, 1.8392868, etc. The n-th convergents are 2 (n=1), 11/6 (n=2), 46/25 (n=3), 103/56 (n=4), 31451/17105 (n=5) etc., each with associated rounded decimal expansions.
%C A138371 a(n) is the maximum number of post-period digits of the two expansions 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 A138371 For n=4, the 4th convergent is 103/56 = 1.83928571..., with a sequence of rounded representations 2, 1.8, 1.84, 1.839, 1.8393, 1.83929, 1.839286, 1.8392857 etc.
%e A138371 Rounded to 1, 2, 3, 4 or 5 post-period decimal digits, this is the same as the rounded version of the exact value, but disagrees if both are rounded to 6 decimal digits, where 1.839287 <> 1.839286.
%e A138371 So a(4) = 5 (digits), the maximum rounding level with agreement.
%Y A138371 Cf. A138335, A138336, A138337, A138339, A138343, A138366, A138367, A138369, A138370.
%K A138371 base,nonn
%O A138371 1,3
%A A138371 _Artur Jasinski_, Mar 17 2008
%E A138371 Definition and values replaced as defined via continued fractions by _R. J. Mathar_, Oct 01 2009