cp's OEIS Frontend

A188859 Decimal expansion of 2 - log(4).

%I A188859 #29 Apr 06 2025 16:52:22
%S A188859 6,1,3,7,0,5,6,3,8,8,8,0,1,0,9,3,8,1,1,6,5,5,3,5,7,5,7,0,8,3,6,4,6,8,
%T A188859 6,3,8,4,8,9,9,9,7,3,1,2,7,9,4,8,9,4,9,1,7,5,8,6,3,9,9,8,1,0,1,3,2,1,
%U A188859 2,7,5,6,0,6,0,6,1,0,5,6,8,7,8,8,2,7,3,3,4,6,0,0,7,1,6,2,6,2,4,9,1,5,9,9,7
%N A188859 Decimal expansion of 2 - log(4).
%C A188859 Limit as n increases without bound of the probability that n mod m is less than m/2, with m chosen uniformly at random from 1..n. (As usual, 0 <= n mod m < m.)
%H A188859 Jean-Paul Allouche and Jeffrey Shallit, <a href="https://doi.org/10.1007/BFb0097122">Sums of digits and the Hurwitz zeta function</a>, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.
%H A188859 I. S. Gradsteyn and I. M. Ryzhik, <a href="http://mathtable.com/gr/index.html">Table of integrals, series and products</a> (6th ed.), 2000, (eq. 0.236.5).
%H A188859 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A188859 From _Amiram Eldar_, Aug 15 2020: (Start)
%F A188859 Equals Sum_{k>=1} 1/(2*k^2 + k).
%F A188859 Equals -Integral_{x=0..1} log(1-x^2) dx. (End)
%F A188859 Equals Sum_{k>=1} A023416(k)/(k*(k+1)) (Allouche and Shallit, 1990). - _Amiram Eldar_, Jun 01 2021
%F A188859 Equals 1/(1 + 2/(3 + 1^2/(4 + 3^2/(5 + 2^2/(6 + 4^2/(7 + 3^2/(8 + 5^2/(9 + 4^2/(10 + 6^2/(11 + ... + (n-1)^2/((2*n) + (n+1)^2/((2*n+1) + ... )))))))))))). Cf. A016639. - _Peter Bala_, Mar 04 2024
%F A188859 Equals 1/2 + Sum_{k>=1} 1/(k*(4*k^2-1)^2). - _Sean A. Irvine_, Apr 06 2025
%e A188859 0.61370563888010938116553575708364686384899973127949...
%t A188859 RealDigits[2 - Log[4], 10, 120][[1]]
%o A188859 (PARI) vecextract(eval(Vec(Str(2-log(4)))),"3..")
%Y A188859 Cf. A016639, A023416, A244009.
%K A188859 nonn,cons,easy
%O A188859 0,1
%A A188859 _Charles R Greathouse IV_, Apr 12 2011