A305879 Number of binary places to which n-th convergent of continued fraction expansion of Pi matches the correct value.

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

Offset

1,1

Comments

For the similar case of number of correct decimal places see A084407.

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), 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