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
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
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