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

0, -1, 3, 6, 9, 13, 14, 17, 19, 20, 23, 20, 25, 20, 33, 37, 35, 38, 41, 43, 45, 43, 47, 48, 52, 54, 58, 61, 68, 70, 74, 77, 78, 81, 86, 89, 92, 93, 92, 99, 105, 109, 113, 116, 118, 121, 127, 133, 136, 135, 139, 141, 145, 149, 154, 159, 161, 165, 171, 173, 172, 180

Offset

1,3

Comments

Binary expansion of log(2) in A068426.

For number of correct decimal digits see A317558.

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

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.

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).

Links

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Formula

Lim_{n -> oo} a(n)/n = 2*log(A086702)/log(2) = 2*A100199/log(2) = 2*A305607.

Example

   n   convergent         binary expansion         a(n)
  ==  ============  =============================  ====
   1     0 / 1      0.0                              0
   2     1 / 1      1.0                             -1
   3     2 / 3      0.1010...                        3
   4     7 / 10     0.1011001...                     6
   5     9 / 13     0.1011000100...                  9
   6    61 / 88     0.10110001011101...             13
   7   192 / 277    0.101100010111000...            14
   8   253 / 365    0.101100010111001001...         17
   9   445 / 642    0.10110001011100100000...       19
  10  1143 / 1649   0.101100010111001000011...      20
  oo  lim = log(2)  0.101100010111001000010111...   --

Mathematica

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 *)

Crossrefs

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

Sequence in context: A065811 A061514 A078559 * A338763 A159908 A236761

Adjacent sequences: A317554 A317555 A317556 * A317558 A317559 A317560

Keyword

sign,base

Author

A.H.M. Smeets, Jul 31 2018

Extensions

a(40) onward from Robert G. Wilson v, Aug 09 2018

Status

approved