A386735 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} {1/(x+y)}^2 dx dy, where {} denotes fractional part.

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, 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, 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

Offset

0,1

Links

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

Ovidiu Furdui, Exotic fractional part integrals and Euler's constant, Analysis, Vol. 31 (2011), pp. 249-257.

Huizeng Qin and Youmin Lu, Integrals of Fractional Parts and Some New Identities on Bernoulli Numbers, 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.

Formula

Equals 5/2 - log(2) - gamma - Pi^2/12.

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).

Example

0.40717012111440861174004820513640840627286557909642...

Mathematica

RealDigits[5/2 - Log[2] - EulerGamma - Pi^2/12, 10, 120][[1]]

Prog

(PARI) 5/2 - log(2) - Euler - Pi^2/12

Crossrefs

Cf. A001620 (gamma), A002162, A072691, A386733 (m=1).

Sequence in context: A199071 A351898 A157698 * A342360 A251967 A330725

Adjacent sequences: A386732 A386733 A386734 * A386736 A386737 A386738

Keyword

nonn,cons

Author

Amiram Eldar, Aug 01 2025

Status

approved