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A274400 Decimal expansion of 'V', the value of a 4-dimensional iterated integral studied by David Broadhurst in connection with Quantum Field Theory (negated). 3
0, 3, 9, 0, 1, 2, 7, 2, 6, 3, 6, 0, 1, 6, 7, 1, 6, 6, 0, 1, 7, 5, 6, 6, 9, 4, 1, 3, 5, 4, 5, 0, 0, 4, 6, 1, 7, 5, 7, 4, 7, 5, 8, 5, 7, 1, 3, 8, 6, 1, 3, 0, 9, 9, 0, 1, 4, 9, 3, 8, 9, 6, 7, 3, 9, 5, 4, 0, 3, 8, 9, 2, 7, 5, 0, 1, 8, 5, 6, 5, 4, 8, 7, 1, 8, 1, 2, 1, 8, 8, 1, 2, 8, 2, 8, 4, 2, 6, 1, 2, 8, 8 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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=0..101.

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;

FORMULA

V = Sum_{j>k>0} (-1)^j cos(2Pi k/3)/(j^3 k).

Equals 3 zeta(3)/8-1/2+Sum_{k>=2} ((-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)).

EXAMPLE

-0.0390127263601671660175669413545004617574758571386130990149389673954...

MATHEMATICA

digits = 101;

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 = 3 Zeta[3]/8 - 1/2 + NSum[v[k], {k, 2, Infinity}, WorkingPrecision -> digits + 10, Method -> "AlternatingSigns"]; Join[{0},

RealDigits[V, 10, digits][[1]]]

CROSSREFS

Cf. A255685 ('U' in Borwein & Borwein).

Sequence in context: A088110 A122759 A247042 * A200495 A272535 A016626

Adjacent sequences:  A274397 A274398 A274399 * A274401 A274402 A274403

KEYWORD

nonn,cons

AUTHOR

Jean-Fran├žois Alcover, Jun 20 2016

STATUS

approved

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Last modified January 15 19:35 EST 2019. Contains 319171 sequences. (Running on oeis4.)