A274420 Decimal expansion of V_5, a Quantum Field Theory constant [negated] related to the coloring of the tetrahedron with five masses.

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

Offset

1,1

References

Jonathan Borwein and Peter Borwein, Experimental and Computational Mathematics: Selected Writings, Perfectly Scientific Press, 2010, p. 106.

Links

Table of n, a(n) for n=1..101.

David J. Broadhurst, Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity, arXiv:hep-th/9803091, 1998; p. 18.

Formula

V_5 = 6 zeta(3) - 469/27 zeta(4) + 8/3 C^2 - 16 V, where C is A143298 and V A274400.

Example

-8.21685981750873806291339833860108582496950833917257503683557579115...

Mathematica

digits = 101;
C0 = A143298 = (9 - PolyGamma[1, 2/3] + PolyGamma[1, 4/3])/(4*Sqrt[3]);
v[k_] := ((-1)^k*((24*(k - 1)*(3*k - 4))/(3*k - 2)^3 + (8*(3*k*(3*k - 5) + 4))/(27*(k - 1)^3) + PolyGamma[2, (3*k)/2 - 1] - PolyGamma[2, (3*(k - 1))/2]))/(48*(k - 1)*(3*k - 4)*(3*k - 2));
V = A274400 = 3 Zeta[3]/8 - 1/2 + NSum[v[k], {k, 2, Infinity}, WorkingPrecision -> digits + 10, Method -> "AlternatingSigns"];
V5 = 6 Zeta[3] - 469/27 Zeta[4] + 8/3 C0^2 - 16 V;
RealDigits[V5, 10, digits][[1]]

Crossrefs

Cf. A274412 (V_1), A274413 (V_2A), A274414 (V_2N), A274415 (V_3T), A274416 (V_3S), A274417 (V_3L), A274418 (V_4A), A274419 (V_4N), A274421 (V_6).

Sequence in context: A157472 A346834 A179640 * A147868 A073442 A177428

Adjacent sequences: A274417 A274418 A274419 * A274421 A274422 A274423

Keyword

nonn,cons

Author

Jean-François Alcover, Jun 21 2016

Status

approved