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 A386564 #12 Jul 26 2025 08:01:51
%S A386564 0,0,7,7,8,8,9,5,5,0,8,4,0,9,6,6,5,2,0,5,4,2,8,3,6,0,9,6,5,9,9,2,7,1,
%T A386564 4,1,1,9,0,1,7,1,9,6,4,8,9,2,6,6,3,2,0,8,4,1,9,1,0,2,4,4,6,9,5,8,0,0,
%U A386564 5,3,5,9,8,6,8,2,9,2,3,4,1,2,0,4,2,2,4,9,6,9,2,9,8,5,4,8,5,7,6,5,9,9,1,7,6
%N A386564 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} ({x/y}*{y/z}*{z/x})^3 dx dy dz, where {w} is the fractional part of w.
%H A386564 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 A386564 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), section 1.48 The Calculation of a Beautiful Triple Fractional Part Integral with a Cubic Power, p. 31.
%F A386564 Equal 1 - 3*(zeta(2)+zeta(3)+zeta(4))/8 + 21*zeta(6)/320 + 7*zeta(8)/160 + zeta(3)^2/40 + zeta(2)*zeta(3)/40 + zeta(2)*zeta(5)/20 + zeta(3)*zeta(4)/16 + zeta(3)*zeta(5)/20 + zeta(4)*zeta(5)/20.
%F A386564 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 A386564 0.00778895508409665205428360965992714119017196489266...
%t A386564 RealDigits[1 - 3*(Zeta[2]+Zeta[3]+Zeta[4])/8 + 21*Zeta[6]/320 + 7*Zeta[8]/160 + Zeta[3]^2/40 + Zeta[2]*Zeta[3]/40 + Zeta[2]*Zeta[5]/20 + Zeta[3]*Zeta[4]/16 + Zeta[3]*Zeta[5]/20 + Zeta[4]*Zeta[5]/20, 10, 120, -1][[1]]
%o A386564 (PARI) 1 - 3*(zeta(2)+zeta(3)+zeta(4))/8 + 21*zeta(6)/320 + 7*zeta(8)/160 + zeta(3)^2/40 + zeta(2)*zeta(3)/40 + zeta(2)*zeta(5)/20 + zeta(3)*zeta(4)/16 + zeta(3)*zeta(5)/20 + zeta(4)*zeta(5)/20
%Y A386564 Cf. A002117, A013661, A013662, A013663, A013664, A013666, A183699, A183700.
%Y A386564 Cf. A375901 (m=1), A383289 (m=2), this constant (m=3).
%K A386564 nonn,cons
%O A386564 0,3
%A A386564 _Amiram Eldar_, Jul 26 2025