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 A386735 #6 Aug 01 2025 11:08:32
%S A386735 4,0,7,1,7,0,1,2,1,1,1,4,4,0,8,6,1,1,7,4,0,0,4,8,2,0,5,1,3,6,4,0,8,4,
%T A386735 0,6,2,7,2,8,6,5,5,7,9,0,9,6,4,2,1,9,2,8,2,0,5,7,7,3,6,4,0,9,3,6,7,3,
%U A386735 4,9,1,6,0,5,1,0,4,0,1,7,6,5,4,0,3,7,5,1,5,9,4,0,1,9,5,5,2,1,0,2,9,1,3,6,4
%N A386735 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} {1/(x+y)}^2 dx dy, where {} denotes fractional part.
%H A386735 Ovidiu Furdui, <a href="http://doi.org/10.1524/anly.2011.1131">Exotic fractional part integrals and Euler's constant</a>, Analysis, Vol. 31 (2011), pp. 249-257.
%H A386735 Huizeng Qin and Youmin Lu, <a href="https://m-hikari.com/ijcms-2011/13-16-2011/luyouminIJCMS13-16-2011.pdf">Integrals of Fractional Parts and Some New Identities on Bernoulli Numbers</a>, Int. J. Contemp. Math. Sciences, Vol. 6, No. 15 (2011), pp. 745-761. See eq. (3.1). Note that this equation has an error, 3/2 instead of 5/2.
%F A386735 Equals 5/2 - log(2) - gamma - Pi^2/12.
%F A386735 For m >= 3, Integral_{x=0..1} Integral_{y=0..1} {1/(x+y)}^m dx dy = (2^(2-m) + m - 3)/((m-1)*(m-2)) + (m!/2) * Sum_{j>=1} ((j+1)!/(m+j)!) * (zeta(j+2) - 1).
%e A386735 0.40717012111440861174004820513640840627286557909642...
%t A386735 RealDigits[5/2 - Log[2] - EulerGamma - Pi^2/12, 10, 120][[1]]
%o A386735 (PARI) 5/2 - log(2) - Euler - Pi^2/12
%Y A386735 Cf. A001620 (gamma), A002162, A072691, A386733 (m=1).
%K A386735 nonn,cons
%O A386735 0,1
%A A386735 _Amiram Eldar_, Aug 01 2025