A305607 - OEIS

%I #25 Jul 12 2018 00:51:44

%S 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,

%T 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,

%U 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

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

%C 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.

%C 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.

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

%C 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.

%F Equals A100199/log(2).

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

%e 1.71185737126865169874676283878247783620154351162446786...

%t RealDigits[(Pi/Log@2)^2/12, 10, 111][[1]] (* _Robert G. Wilson v_, Jun 13 2018 *)

%o (PARI) (Pi/log(2))^2/12 \\ _Michel Marcus_, Jul 03 2018

%Y Cf. A100199, A240995.

%K nonn,cons,easy

%O 1,2

%A _A.H.M. Smeets_, Jun 05 2018