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

Author

A.H.M. Smeets, Jul 31 2018

Keywords

• sign
• base

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.

Examples

			   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...   --
		

Links

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

Crossrefs

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

Programs

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 *)

Formula

Lim_{n -> oo} a(n)/n = 2*log(A086702)/log(10) = 2*A100199/log(10) = 2*A240995.

Extensions

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