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A317557 Number of binary digits to which the n-th convergent of the continued fraction expansion of log(2) matches the correct value. 3

%I

%S 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 52,54,58,61,68,70,74,77,78,81,86,89,92,93,92,99,105,109,113,116,118,

%U 121,127,133,136,135,139,141,145,149,154,159,161,165,171,173,172,180

%N Number of binary digits to which the n-th convergent of the continued fraction expansion of log(2) matches the correct value.

%C Binary expansion of log(2) in A068426.

%C For number of correct decimal digits see A317558.

%C For the similar case of number of correct binary digits of Pi see A305879.

%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), and the sequence for hexadecimal digits is obtained by floor(a(n)/4).

%H A.H.M. Smeets, <a href="/A317557/b317557.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 n convergent binary expansion a(n)

%e == ============ ============================= ====

%e 1 0 / 1 0.0 0

%e 2 1 / 1 1.0 -1

%e 3 2 / 3 0.1010... 3

%e 4 7 / 10 0.1011001... 6

%e 5 9 / 13 0.1011000100... 9

%e 6 61 / 88 0.10110001011101... 13

%e 7 192 / 277 0.101100010111000... 14

%e 8 253 / 365 0.101100010111001001... 17

%e 9 445 / 642 0.10110001011100100000... 19

%e 10 1143 / 1649 0.101100010111001000011... 20

%e oo lim = log(2) 0.101100010111001000010111... --

%t 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 Cf. A016730, A068426, A086702, A100199, A305607, A317558.

%K sign,base

%O 1,3

%A _A.H.M. Smeets_, Jul 31 2018

%E a(40) onward from _Robert G. Wilson v_, Aug 09 2018

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Last modified April 5 13:26 EDT 2020. Contains 333241 sequences. (Running on oeis4.)