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%I #34 Nov 21 2019 00:11:04
%S 2,8,13,21,28,31,28,34,32,38,40,44,47,51,52,54,57,60,62,64,70,78,80,
%T 81,84,91,94,100,103,104,107,116,121,132,133,136,133,144,148,152,148,
%U 156,158,165,167,170,173,176,179,182
%N Number of binary places to which n-th convergent of continued fraction expansion of Pi matches the correct value.
%C For the similar case of number of correct decimal places see A084407.
%C The denominator of the k-th convergent obtained from a continued fraction satisfying the Gauss-Kuzmin distribution will tend to exp(k*A100199), A100199 being the inverse of Lévy's constant; the error between the k-th convergent and the constant itself tends to exp(-2*k*A100199), or in binary digits 2*k*A100199/log(2) bits after the binary point.
%C The sequence for quaternary digits is obtained by floor(a(n)/2), the sequence for octal digits is obtained by floor(a(n)/3), the sequence for hexadecimal digits is obtained by floor(a(n)/4).
%H A.H.M. Smeets, <a href="/A305879/b305879.txt">Table of n, a(n) for n = 1..20000</a>
%F Lim {n -> oo} (a(n)/n) = 2*log(A086702)/log(2) = 2*A100199/log(2) = 2*A305607.
%e Pi = 11.0010010000111111...
%e n=1: 3/1 = 11.000... so a(1) = 2
%e n=2: 22/7 = 11.001001001... so a(2) = 8
%e n=3: 333/106 = 11.00100100001110... so a(3) = 13
%Y Cf. A084407, A086702, A100199, A305607.
%K nonn,base
%O 1,1
%A _A.H.M. Smeets_, Jun 13 2018
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