A386564 - OEIS

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.

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, 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, 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

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OFFSET

0,3

LINKS

Table of n, a(n) for n=0..104.

Cornel Ioan Vălean, Problem 11902, Problems and Solutions, The American Mathematical Monthly, Vol. 123, No. 4 (2016), p. 399; A Row of Zetas, Solution to Problem 11902 by Rituraj Nandan, ibid., Vol. 125, No. 2 (2018), pp. 182-184.

Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer (2019), section 1.48 The Calculation of a Beautiful Triple Fractional Part Integral with a Cubic Power, p. 31.

FORMULA

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.

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)).

EXAMPLE

0.00778895508409665205428360965992714119017196489266...

MATHEMATICA

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]]

PROG

(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

CROSSREFS

Cf. A002117, A013661, A013662, A013663, A013664, A013666, A183699, A183700.

Cf. A375901 (m=1), A383289 (m=2), this constant (m=3).

Sequence in context: A294968 A096251 A108186 * A024818 A076236 A072520

Adjacent sequences: A386561 A386562 A386563 * A386565 A386566 A386567

KEYWORD

nonn,cons

AUTHOR

Amiram Eldar, Jul 26 2025

STATUS

approved