A317558 Number of decimal digits to which the n-th convergent of the continued fraction expansion of log(2) matches the correct value.

0, -1, 1, 0, 2, 4, 5, 4, 5, 6, 6, 6, 7, 8, 9, 10, 11, 10, 12, 13, 13, 13, 14, 15, 15, 16, 17, 18, 20, 22, 22, 23, 23, 24, 25, 26, 27, 27, 28, 29, 31, 32, 33, 34, 35, 36, 38, 40, 39, 41, 39, 43, 44, 45, 46, 48, 48, 49, 51, 52, 52, 54, 54, 55, 55, 56, 57, 57, 58

Offset

1,5

Comments

Decimal expansion of log(2) in A002162.

For the number of correct binary digits see A317557.

For the similar case of number of correct decimal digits of Pi see A084407.

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(10) = 2*A100199/log(10) = 2*A240995.

Example

   n   convergent    decimal expansion    a(n)
  ==  ============  ====================  ====
   1     0 / 1      0.0                     0
   2     1 / 1      1.0                    -1
   3     2 / 3      0.66...                 1
   4     7 / 10     0.7...                  0
   5     9 / 13     0.692...                2
   6    61 / 88     0.69318...              4
   7   192 / 277    0.693140...             5
   8   253 / 365    0.69315...              4
   9   445 / 642    0.693146...             5
  10  1143 / 1649   0.6931473...            6
  oo  lim = log(2)  0.693147180559945...   --

Mathematica

a[n_] := Block[{k = 1, a = RealDigits[Log@2, 10, n + 10][[1]], b = RealDigits[ FromContinuedFraction@ ContinuedFraction[ Log@2, n], 10, n + 10][[1]]}, While[a[[k]] == b[[k]], k++]; k - 1]; a[1] = 0; a[2] = -1; Array[a, 69] (* Robert G. Wilson v, Aug 09 2018 *)

Crossrefs

Cf. A002162, A016730, A086702, A100199, A240995, A317557.

Sequence in context: A317499 A004581 A212790 * A070784 A200289 A165044

Adjacent sequences: A317555 A317556 A317557 * A317559 A317560 A317561

Keyword

sign,base

Author

A.H.M. Smeets, Jul 31 2018

Extensions

a(61) onward from Robert G. Wilson v, Aug 09 2018

Status

approved