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

Author

A.H.M. Smeets, Jun 13 2018

Keywords

• nonn
• base

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).

Examples

			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
		

Links

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

Crossrefs

Cf. A084407, A086702, A100199, A305607.

Formula

Lim {n -> oo} (a(n)/n) = 2*log(A086702)/log(2) = 2*A100199/log(2) = 2*A305607.