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A084407 Number of decimal places to which the n-th convergent of continued fraction expansion of Pi matches with the correct value.

%I A084407 #26 Jan 27 2022 21:59:55
%S A084407 0,2,4,6,9,9,9,9,11,10,12,12,14,15,15,16,17,17,18,19,21,23,24,24,25,
%T A084407 27,29,30,30,32,33,34,37,39,40,40,41,42,44,45,45,46,47,49,50,51,51,53,
%U A084407 54,55,55,56,56,58,59,59,60,60,61,60,62,64,63,64,65,65,67,67,68,70,69,71
%N A084407 Number of decimal places to which the n-th convergent of continued fraction expansion of Pi matches with the correct value.
%C A084407 The n-th convergent of the continued fraction expansion of Pi is A002485(n+1)/A002486(n+1).
%H A084407 A.H.M. Smeets, <a href="/A084407/b084407.txt">Table of n, a(n) for n = 1..20000</a>
%H A084407 F. Richman, <a href="http://math.fau.edu/Richman/contfrac.htm">Continued fractions</a>
%F A084407 Limit_{n -> oo} a(n)/n = 2*log(A086702)/log(10) = 2*A100199/log(10) = 2*A240995. -  _A.H.M. Smeets_, Jun 13 2018
%e A084407 From _A.H.M. Smeets_, Jun 13 2018: (Start)
%e A084407 Pi = 3.141592653589...
%e A084407 n=1: 3/1 = 3.0... so a(1) = 0;
%e A084407 n=2: 22/7 = 3.142... so a(2) = 2;
%e A084407 n=3: 333/106 = 3.14150... so a(3) = 4;
%e A084407 n=4: 355/113 = 3.1415929... so a(4) = 6;
%e A084407 n=5: 103993/33102 = 3.1415926530... so a(5) = 9;
%e A084407 n=6: 104348/33215 = 3.1415926539... so a(6) = 9;
%e A084407 n=7: 208341/66317 = 3.1415926534... so a(7) = 9;
%e A084407 n=8: 312689/99532 = 3.1415926536... so a(8) = 9;
%e A084407 n=9: 833719/265381 = 3.141592653581... so a(9) = 11;
%e A084407 n=10: 1146408/364913 = 3.14159265359... so a(10) = 10. (End)
%Y A084407 Cf. A000796.
%Y A084407 Cf. A002485, A002486, A114526.
%Y A084407 Cf. A086702, A100199, A240995.
%K A084407 base,nonn
%O A084407 1,2
%A A084407 _Lekraj Beedassy_, Jun 24 2003
%E A084407 More terms from _Vladeta Jovovic_, Jun 27 2003