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 A383289 #24 Jul 26 2025 08:02:40
%S A383289 0,2,3,4,0,9,6,1,8,2,3,1,5,8,0,8,7,2,6,8,0,2,0,0,9,3,8,5,5,0,0,6,9,8,
%T A383289 0,6,7,5,8,4,0,4,4,2,5,8,2,7,1,4,8,3,8,5,1,5,9,3,8,7,1,0,0,9,6,3,8,8,
%U A383289 8,3,3,5,9,5,8,3,1,8,0,5,9,4,1,0,4,1,5,6,4,9,6,6,8,0,3,9,4,0,0,5,3,8,9,4,0,0,1
%N A383289 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} ({x/y}*{y/z}*{z/x})^2 dx dy dz, where {w} is the fractional part of w.
%H A383289 Cornel Ioan Vălean, <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.123.4.399">Problem 11902</a>, Problems and Solutions, The American Mathematical Monthly, Vol. 123, No. 4 (2016), p. 399; <a href="https://www.jstor.org/stable/48661846">A Row of Zetas</a>, Solution to Problem 11902 by Rituraj Nandan, ibid., Vol. 125, No. 2 (2018), pp. 182-184.
%H A383289 Cornel Ioan Vălean, <a href="https://ia601603.us.archive.org/20/items/1.pdf_almost/1.pdf.pdf">(Almost) Impossible Integrals, Sums, and Series</a>, Springer (2019), p. viii.
%F A383289 Equals 1 - zeta(2)/2 - zeta(3)/2 + 7*zeta(6)/48 + zeta(2)*zeta(3)/18 + zeta(3)^2/18 + zeta(3)*zeta(4)/12.
%F A383289 In general, Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} ({x/y}*{y/z}*{z/x})^m dx dy dz = 1 - 3*Sum_{j=1..m} zeta(j+1)/(2*(m+1)) + (Sum_{j=1..m} zeta(j+1))*(Sum_{j=1..m} (j+1)*zeta(j+2))/((m+1)^2*(m+2)).
%e A383289 0.02340961823158087268020093855006980675840442582714...
%t A383289 RealDigits[1 - Zeta[2]/2 - Zeta[3]/2 + 7*Zeta[6]/48 + Zeta[2]*Zeta[3]/18 + Zeta[3]^2/18 + Zeta[3]*Zeta[4]/12, 10, 120, -1][[1]]
%t A383289 RealDigits[With[{m = 2}, 1 - 3*Sum[Zeta[j + 1], {j, 1, m}]/(2*(m + 1)) + Sum[Zeta[j + 1], {j, 1, m}] * Sum[(j + 1)*Zeta[j + 2], {j, 1, m}]/((m + 1)^2*(m + 2))], 10, 106][[1]] (* _Vaclav Kotesovec_, Jul 26 2025, following the general formula found by the solvers *)
%o A383289 (PARI) 1 - zeta(2)/2 - zeta(3)/2 + 7*zeta(6)/48 + zeta(2)*zeta(3)/18 + zeta(3)^2/18 + zeta(3)*zeta(4)/12
%Y A383289 Cf. A002117, A013661, A013662, A013664, A183699, A183700.
%Y A383289 Cf. A375901 (m=1), this constant (m=2), A386564 (m=3).
%K A383289 nonn,cons,nice
%O A383289 0,2
%A A383289 _Amiram Eldar_, Jul 26 2025