A386714 Decimal expansion of Integral_{x=0..1} {1/x}^3 * {1/(1-x)}^3 dx, where {} denotes fractional part.

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

Offset

0,3

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 -zeta(2) + 3*gamma + 36*log(A) - 6*log(2*Pi) + 2, where gamma is Euler's constant and A is the Glaisher-Kinkelin constant.

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

0.01461170261653269568427203547387356507606811502683...

Mathematica

RealDigits[-Zeta[2] + 3*EulerGamma + 36*Log[Glaisher] - 6*Log[2*Pi] + 2, 10, 120, -1][[1]]

Prog

(PARI) -zeta(2) + 3*Euler + 36*(1/12-zeta'(-1)) - 6*log(2*Pi) + 2

Crossrefs

Cf. A001620 (gamma), A013661, A061444, A074962 (A), A225746.

Cf. A147533 (m=1), A386713 (m=2), this constant (m=3).

Sequence in context: A107951 A019646 A238582 * A396444 A395934 A384213

Adjacent sequences: A386711 A386712 A386713 * A386715 A386716 A386717

Keyword

nonn,cons

Author

Amiram Eldar, Jul 31 2025

Status

approved