A305607 Decimal expansion of (Pi/log(2))^2/12.

1, 7, 1, 1, 8, 5, 7, 3, 7, 1, 2, 6, 8, 6, 5, 1, 6, 9, 8, 7, 4, 6, 7, 6, 2, 8, 3, 8, 7, 8, 2, 4, 7, 7, 8, 3, 6, 2, 0, 1, 5, 4, 3, 5, 1, 1, 6, 2, 4, 4, 6, 7, 8, 6, 3, 6, 4, 2, 0, 8, 7, 3, 3, 0, 2, 1, 1, 0, 7, 6, 0, 8, 4, 9, 6, 1, 8, 6, 9, 7, 8, 2, 6, 2, 0, 2, 6, 9, 5, 9, 2, 7, 4, 5, 2, 3, 0, 3, 9, 4, 4

Offset: 1

Author

A.H.M. Smeets, Jun 05 2018

Keywords

• nonn
• cons
• easy

Comments

The constant represents the mean information density per continued fraction term for continued fraction terms satisfying the Gauss-Kuzmin distribution in bits per term, i.e., for a finite continued fraction (fractional, n/d), the denominator d has approximately (1/12)*(Pi/log(2))^2*t binary digits are obtained correctly, where t is the number of terms.

For infinite continued fractions satisfying Gauss-Kuzmin distribution, about 2*(1/12)*(Pi/log(2))^2*t binary digits are obtained correctly from the first t continued fraction terms.

Note that A240995 represents the mean information density in decimal digits per term.

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; i.e., in binary digits, the k-th convergent tends to A100199/log(2) binary digits.

Examples

			1.71185737126865169874676283878247783620154351162446786...
		

Crossrefs

Cf. A100199, A240995.

Programs

Mathematica

RealDigits[(Pi/Log@2)^2/12, 10, 111][[1]] (* Robert G. Wilson v, Jun 13 2018 *)

PARI

(Pi/log(2))^2/12 \\ Michel Marcus, Jul 03 2018

Formula

Equals A100199/log(2).

Equals A240995*log(10)/log(2).