A249652 Decimal expansion of integral_{0..1} Li_3(x)^2 dx, where Li_3 is the trilogarithm function.

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

Offset

0,1

Comments

In the article by Freitas, the term +(15/2)*zeta(4) is incorrectly given a negative sign. - Vaclav Kotesovec, Jun 30 2026

Links

Table of n, a(n) for n=0..101.

Pedro Freitas, Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums, Math. Comput., Vol. 74, No. 251 (2005), pp. 1425-1440. See p. 1435, Table 3.

Eric Weisstein's MathWorld, Trilogarithm.

Formula

Equals 20 - 8*zeta(2) - 10*zeta(3) + (15/2)*zeta(4) - 2*zeta(2)*zeta(3) + zeta(3)^2.

Example

0.4277147842908240881128389716127945324286...

Mathematica

RealDigits[20 - 8*Zeta[2] - 10*Zeta[3] + (15/2)*Zeta[4] - 2*Zeta[2]*Zeta[3] + Zeta[3]^2, 10, 102] // First
(* Alternative: *)
NIntegrate[PolyLog[3, x]^2, {x, 0, 1}, WorkingPrecision->102] (* Vaclav Kotesovec, Nov 03 2014 *)

Prog

(PARI) z2=zeta(2); z3=zeta(3); 20 - 8*z2 - 10*z3 + 15*zeta(4)/2 - 2*z2*z3 + z3^2 \\ Charles R Greathouse IV, Apr 20 2016
(Python)
from mpmath import mp, zeta
mp.dps=103
z2=zeta(2)
z3=zeta(3)
print([int(z) for z in list(str(20 - 8*z2 - 10*z3 + 15*zeta(4)/2 - 2*z2*z3 + z3**2)[2:-1])]) # Indranil Ghosh, Jul 03 2017

Crossrefs

Cf. A002117, A013661, A013662, A249649, A249651, A397550, A397555.

Sequence in context: A092314 A237750 A372860 * A110841 A380547 A128226

Adjacent sequences: A249649 A249650 A249651 * A249653 A249654 A249655

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Nov 03 2014

Status

approved