This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A317557 #29 Nov 21 2019 00:11:24
%S A317557 0,-1,3,6,9,13,14,17,19,20,23,20,25,20,33,37,35,38,41,43,45,43,47,48,
%T A317557 52,54,58,61,68,70,74,77,78,81,86,89,92,93,92,99,105,109,113,116,118,
%U A317557 121,127,133,136,135,139,141,145,149,154,159,161,165,171,173,172,180
%N A317557 Number of binary digits to which the n-th convergent of the continued fraction expansion of log(2) matches the correct value.
%C A317557 Binary expansion of log(2) in A068426.
%C A317557 For number of correct decimal digits see A317558.
%C A317557 For the similar case of number of correct binary digits of Pi see A305879.
%C A317557 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 A317557 The sequence for quaternary digits is obtained by floor(a(n)/2), the sequence for octal digits is obtained by floor(a(n)/3), and the sequence for hexadecimal digits is obtained by floor(a(n)/4).
%H A317557 A.H.M. Smeets, <a href="/A317557/b317557.txt">Table of n, a(n) for n = 1..20000</a>
%F A317557 Lim_{n -> oo} a(n)/n = 2*log(A086702)/log(2) = 2*A100199/log(2) = 2*A305607.
%e A317557 n convergent binary expansion a(n)
%e A317557 == ============ ============================= ====
%e A317557 1 0 / 1 0.0 0
%e A317557 2 1 / 1 1.0 -1
%e A317557 3 2 / 3 0.1010... 3
%e A317557 4 7 / 10 0.1011001... 6
%e A317557 5 9 / 13 0.1011000100... 9
%e A317557 6 61 / 88 0.10110001011101... 13
%e A317557 7 192 / 277 0.101100010111000... 14
%e A317557 8 253 / 365 0.101100010111001001... 17
%e A317557 9 445 / 642 0.10110001011100100000... 19
%e A317557 10 1143 / 1649 0.101100010111001000011... 20
%e A317557 oo lim = log(2) 0.101100010111001000010111... --
%t A317557 a[n_] := Block[{k = 1, a = RealDigits[ Log@2, 2, 4 + 10][[1]], b = RealDigits[ FromContinuedFraction@ ContinuedFraction[Log@2, n + 1], 2, 4n + 10][[1]]}, While[ a[[k]] == b[[k]], k++]; k - 1]; a[1] = 0; a[2] = -1; Array[a, 61] (* _Robert G. Wilson v_, Aug 09 2018 *)
%Y A317557 Cf. A016730, A068426, A086702, A100199, A305607, A317558.
%K A317557 sign,base
%O A317557 1,3
%A A317557 _A.H.M. Smeets_, Jul 31 2018
%E A317557 a(40) onward from _Robert G. Wilson v_, Aug 09 2018