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A305879 Number of binary places to which n-th 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 (list; graph; refs; listen; history; text; internal format)
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 quarternary 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

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Last modified October 15 00:14 EDT 2019. Contains 328025 sequences. (Running on oeis4.)