A386713 Decimal expansion of Integral_{x=0..1} {1/x}^2 * {1/(1-x)}^2 dx, where {} denotes fractional part.

0, 4, 2, 6, 4, 5, 6, 0, 6, 0, 3, 1, 2, 5, 0, 4, 9, 1, 8, 1, 6, 5, 8, 9, 5, 3, 0, 9, 1, 5, 3, 3, 1, 3, 9, 4, 7, 2, 2, 5, 4, 2, 4, 4, 5, 3, 4, 2, 5, 7, 2, 9, 0, 7, 3, 1, 4, 1, 4, 3, 3, 8, 4, 3, 2, 2, 6, 5, 4, 6, 6, 0, 3, 0, 7, 4, 2, 4, 4, 9, 7, 8, 1, 0, 1, 5, 8, 1, 3, 5, 9, 2, 0, 6, 4, 6, 5, 8, 2, 9, 1, 7, 5, 1, 9

Offset

0,2

References

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See section 2.11, page 101.

Links

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

Ovidiu Furdui, A class of fractional part integrals and zeta function values, Integral Transforms and Special Functions, Vol. 24, No. 6 (2013), pp. 485-490.

Formula

Equals 4*log(2*pi) - 4*gamma - 5.

Equals 4*A345208 - 1.

In general, for m >= 2, Integral_{x=0..1} {1/x}^m * {1/(1-x)}^m dx = 2 * (Sum_{j=2..m-1} (-1)^(m+j-1) * (zeta(j)-1)) + (-1)^m - (2*m) * Sum_{k>=0} (zeta(2*k+m) - zeta(2*k+m+1))/(k+m) (note that the first sum vanishes when m = 2).

Example

0.04264560603125049181658953091533139472254244534257...

Mathematica

RealDigits[4*Log[2*Pi] - 4*EulerGamma - 5, 10, 120, -1][[1]]

Prog

(PARI) 4*log(2*Pi) - 4*Euler - 5

Crossrefs

Cf. A001620 (gamma), A061444, A345208.

Cf. A147533 (m=1), this constant (m=2), A386714 (m=3).

Sequence in context: A246776 A138614 A161912 * A162339 A200697 A395771

Adjacent sequences: A386710 A386711 A386712 * A386714 A386715 A386716

Keyword

nonn,cons

Author

Amiram Eldar, Jul 31 2025

Status

approved