This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A386713 #6 Jul 31 2025 08:41:31
%S A386713 0,4,2,6,4,5,6,0,6,0,3,1,2,5,0,4,9,1,8,1,6,5,8,9,5,3,0,9,1,5,3,3,1,3,
%T A386713 9,4,7,2,2,5,4,2,4,4,5,3,4,2,5,7,2,9,0,7,3,1,4,1,4,3,3,8,4,3,2,2,6,5,
%U A386713 4,6,6,0,3,0,7,4,2,4,4,9,7,8,1,0,1,5,8,1,3,5,9,2,0,6,4,6,5,8,2,9,1,7,5,1,9
%N A386713 Decimal expansion of Integral_{x=0..1} {1/x}^2 * {1/(1-x)}^2 dx, where {} denotes fractional part.
%D A386713 Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See section 2.11, page 101.
%H A386713 Ovidiu Furdui, <a href="https://doi.org/10.1080/10652469.2012.708869">A class of fractional part integrals and zeta function values</a>, Integral Transforms and Special Functions, Vol. 24, No. 6 (2013), pp. 485-490.
%F A386713 Equals 4*log(2*pi) - 4*gamma - 5.
%F A386713 Equals 4*A345208 - 1.
%F A386713 In general, for m >= 2, Integral_{x=0..1} {1/x}^m * {1/(1-x)}^m dx = 2 * (Sum_{j=2..m-1} (-1)^(m+j-1) * (zeta(j)-1)) + (-1)^m - (2*m) * Sum_{k>=0} (zeta(2*k+m) - zeta(2*k+m+1))/(k+m) (note that the first sum vanishes when m = 2).
%e A386713 0.04264560603125049181658953091533139472254244534257...
%t A386713 RealDigits[4*Log[2*Pi] - 4*EulerGamma - 5, 10, 120, -1][[1]]
%o A386713 (PARI) 4*log(2*Pi) - 4*Euler - 5
%Y A386713 Cf. A001620 (gamma), A061444, A345208.
%Y A386713 Cf. A147533 (m=1), this constant (m=2), A386714 (m=3).
%K A386713 nonn,cons
%O A386713 0,2
%A A386713 _Amiram Eldar_, Jul 31 2025