A274402 Decimal expansion of S_2 = Sum_{n>=0} (2n+1)/((3n+1)^2 (3n+2)^2), a constant related to Quantum Field Theory (see the paper by David Broadhurst).

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

Offset

0,1

Links

Table of n, a(n) for n=0..103.

D. J. Broadhurst, Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity, arXiv:hep-th/9803091, 1998;

Eric Weisstein's MathWorld, Polygamma Function.

Wikipedia, Polygamma Function.

Formula

S_2 = (1/27)*(PolyGamma(1, 1/3) - PolyGamma(1, 2/3)).

Also equals 2/3^(3/2) Cl_2(2Pi/3) where Cl_2 is the Clausen function Cl_2(x) = Sum_{n>0} sin(n x)/n^2.

Example

0.2604341376321620989557291432080307854550447788484284734073666876556...

Mathematica

S2 = (1/27)*(PolyGamma[1, 1/3] - PolyGamma[1, 2/3]);
RealDigits[S2, 10, 104][[1]]

Prog

(PARI)
polygamma(n, x) = if (n == 0, psi(x), (-1)^(n+1)*n!*zetahurwitz(n+1, x));
(polygamma(1, 1/3) - polygamma(1, 2/3))/27 \\ Gheorghe Coserea, Sep 30 2018
(PARI)
clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));
2/3^(3/2) * clausen(2, 2*Pi/3) \\ Gheorghe Coserea, Sep 30 2018
(PARI)
sumpos(n=0, (2*n+1)/((3*n+1)^2*(3*n+2)^2)) \\ Gheorghe Coserea, Sep 30 2018
(PARI)
4/81*sumalt(n=0, (-1/27)^n*(9/(6*n+1)^2 - 9/(6*n+2)^2 - 12/(6*n+3)^2 - 3/(6*n+4)^2 + 1/(6*n+5)^2)) \\ Gheorghe Coserea, Sep 30 2018

Crossrefs

Sequence in context: A084897 A021388 A011040 * A115252 A108431 A190144

Adjacent sequences: A274399 A274400 A274401 * A274403 A274404 A274405

Keyword

nonn,cons

Author

Jean-François Alcover, Jun 20 2016

Status

approved