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A138372 Count of post-period decimal digits up to which the rounded n-th convergent to A103814 agrees with the exact value.

%I A138372 #14 Mar 29 2020 16:53:14
%S A138372 1,3,3,4,3,4,5,6,6,7,8,8,9,10,11,12,13,13,14,16,16,18,21,23,23,24,26,
%T A138372 27,27,29,29,30,30,31,32,34,37,36,39,40,41,42,43,43,46,47,48,48,49,50,
%U A138372 50,52,53,54,55,56,56,58,58,59,62,63,65,66,67,68,69,70,70,71,73,73,75
%N A138372 Count of post-period decimal digits up to which the rounded n-th convergent to A103814 agrees with the exact value.
%C A138372 This is a measure of the quality of the n-th convergent to the Pentanacci constant A103814 if the convergent and the exact value are compared rounded to an increasing number of digits. The sequence of rounded values of A103814 is 2, 2.0, 1.97, 1.966, 1.9659, 1.96595, 1.965948, 1.9659482 etc. The n-th convergents are 2 (n=1), 57/29 (n=2), 116/59 (n=3), 173/88 (n=4), 462/235 (n=5) etc, each with associated rounded decimal expansions.
%C A138372 a(n) is the maximum number of post-period digits of the two expansions if compared at the same level of rounding.
%e A138372 For n=5, the 5th convergent is 462/235 = 1.96595744.., with a sequence of rounded representations 2, 2.0, 1.97, 1.966, 1.9660, 1.96596, etc.
%e A138372 Rounded to 1, 2, or 3 post-period decimal digits, this is the same as the rounded version of the exact value, but disagrees if both are rounded to 4 decimal digits, where 1.9659 <> 1.9660.
%e A138372 So a(5) = 3 (digits), the maximum rounding level with agreement.
%Y A138372 Cf. A138335 - A138339, A138343, A138366, A138367, A138369 - A138371, A138373 - A138374.
%K A138372 base,nonn
%O A138372 1,2
%A A138372 _Artur Jasinski_, Mar 17 2008
%E A138372 Definition and values replaced as defined via continued fractions - _R. J. Mathar_, Oct 01 2009