A274417 - OEIS

%I #9 Oct 01 2018 02:08:14

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

%T 2,8,6,1,5,0,7,4,1,7,8,6,4,1,7,3,7,5,6,2,0,0,5,3,6,7,0,8,7,6,0,2,7,7,

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

%N Decimal expansion of V_3L, a Quantum Field Theory constant [negated] related to the coloring of the tetrahedron with three masses (line case).

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

%H David J. Broadhurst, <a href="http://arxiv.org/abs/hep-th/9803091">Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity</a>, arXiv:hep-th/9803091, 1998; p. 18.

%F V_3L = 6 zeta(3) - 15/4 zeta(4) - 6 C^2, where C is A143298.

%e -3.027009493987652019786374701758957286150741786417375620053670876...

%t C0 = A143298 = (9 - PolyGamma[1, 2/3] + PolyGamma[1, 4/3])/(4*Sqrt[3]);

%t V3L = 6 Zeta[3] - 15/4 Zeta[4] - 6 C0^2;

%t RealDigits[V3L, 10, 102][[1]]

%o (PARI)

%o clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));

%o 6*(zeta(3) - (clausen(2, Pi/3)^2 + (zeta(2)/2)^2)) \\ _Gheorghe Coserea_, Sep 30 2018

%Y Cf. A274412 (V_1), A274413 (V_2A), A274414 (V_2N), A274415 (V_3T), A274416 (V_3S), A274418 (V_4A), A274419 (V_4N), A274420 (V_5), A274421 (V_6).

%K nonn,cons

%O 1,1

%A _Jean-François Alcover_, Jun 21 2016