

A305879


Number of binary places to which nth convergent of continued fraction expansion of Pi matches the correct value.


3



2, 8, 13, 21, 28, 31, 28, 34, 32, 38, 40, 44, 47, 51, 52, 54, 57, 60, 62, 64, 70, 78, 80, 81, 84, 91, 94, 100, 103, 104, 107, 116, 121, 132, 133, 136, 133, 144, 148, 152, 148, 156, 158, 165, 167, 170, 173, 176, 179, 182
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OFFSET

1,1


COMMENTS

For the similar case of number of correct decimal places see A084407.
The denominator of the kth convergent obtained from a continued fraction satisfying the GaussKuzmin distribution will tend to exp(k*A100199), A100199 being the inverse of Lévy's constant; the error between the kth 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), 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

Pi = 11.0010010000111111...
n=1: 3/1 = 11.000... so a(1) = 2
n=2: 22/7 = 11.001001001... so a(2) = 8
n=3: 333/106 = 11.00100100001110... so a(3) = 13


CROSSREFS

Cf. A084407, A086702, A100199, A305607.
Sequence in context: A156245 A247783 A096274 * A271383 A193666 A196024
Adjacent sequences: A305876 A305877 A305878 * A305880 A305881 A305882


KEYWORD

nonn,base


AUTHOR

A.H.M. Smeets, Jun 13 2018


STATUS

approved



