A274416 -id:A274416 - cp's OEIS frontend

A274412 Decimal expansion of V_1, a Quantum Field Theory constant related to the coloring of the tetrahedron with one mass.

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

Offset: 2

Jean-François Alcover, Jun 21 2016

			10.459311120090980286946440058692203652914170609799174013802557978...

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

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.

Cf. A274413 (V_2A), A274414 (V_2N), A274415 (V_3T), A274416 (V_3S), A274417 (V_3L), A274418 (V_4A), A274419 (V_4N), A274420 (V_5), A274421 (V_6).

V1 = 6 Zeta[3] + 3 Zeta[4];
RealDigits[V1, 10, 100][[1]]

6*zeta(3) + 3*zeta(4) \\ Charles R Greathouse IV, Feb 04 2025

V_1 = 6 zeta(3) + 3 zeta(4).

A274413 Decimal expansion of V_2A, a Quantum Field Theory constant related to the coloring of the tetrahedron with two masses (adjacent case).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 21 2016

			1.80072525040187475481841048636286043071616299444935875233874825483...

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

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.

Cf. A274412 (V_1), A274414 (V_2N), A274415 (V_3T), A274416 (V_3S), A274417 (V_3L), A274418 (V_4A), A274419 (V_4N), A274420 (V_5), A274421 (V_6).

V2A = 6 Zeta[3] - 5 Zeta[4];
RealDigits[V2A, 10, 101][[1]]

6*zeta(3) - 5*zeta(4) \\ Charles R Greathouse IV, Feb 04 2025

V_2A = 6 zeta(3) - 5 zeta(4).

A274414 Decimal expansion of V_2N, a Quantum Field Theory constant related to the coloring of the tetrahedron with two masses (non-adjacent case).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 21 2016

			1.1202483970392420822725165482242095262757766719790688973922458664984...

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

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.

Cf. A274412 (V_1), A274413 (V_2A), A274415 (V_3T), A274416 (V_3S), A274417 (V_3L), A274418 (V_4A), A274419 (V_4N), A274420 (V_5), A274421 (V_6).

U = A255685 = Pi^4/180 + (Pi^2/12)*Log[2]^2 - (1/12)*Log[2]^4 - 2*PolyLog[4, 1/2];
V2N = 6 Zeta[3] - 13/2 Zeta[4] - 8 U;
RealDigits[V2N, 10, 103][[1]]

V_2N = 6 zeta(3) - 13/2 zeta(4) - 8 U, where U is A255685.

A274415 Decimal expansion of V_3T, a Quantum Field Theory constant [negated] related to the coloring of the tetrahedron with three masses (triangle case).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 21 2016

			-2.528567684442678011245604299801811180382840813225548878393156606946...

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

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.

Cf. A274412 (V_1), A274413 (V_2A), A274414 (V_2N), A274416 (V_3S), A274417 (V_3L), A274418 (V_4A), A274419 (V_4N), A274420 (V_5), A274421 (V_6).

V3T = 6 Zeta[3] - 9 Zeta[4];
RealDigits[V3T, 10, 101][[1]]

V_3T = 6 zeta(3) - 9 zeta(4).

A274417 Decimal expansion of V_3L, a Quantum Field Theory constant [negated] related to the coloring of the tetrahedron with three masses (line case).

Original entry on oeis.org

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, 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, 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

Offset: 1

Jean-François Alcover, Jun 21 2016

			-3.027009493987652019786374701758957286150741786417375620053670876...

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

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.

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).

C0 = A143298 = (9 - PolyGamma[1, 2/3] + PolyGamma[1, 4/3])/(4*Sqrt[3]);
V3L = 6 Zeta[3] - 15/4 Zeta[4] - 6 C0^2;
RealDigits[V3L, 10, 102][[1]]

clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));
6*(zeta(3) - (clausen(2, Pi/3)^2 + (zeta(2)/2)^2)) \\ Gheorghe Coserea, Sep 30 2018

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

A274418 Decimal expansion of V_4A, a Quantum Field Theory constant [negated] related to the coloring of the tetrahedron with four masses (adjacent case).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 21 2016

			-5.9132047838840205304957178925354050268834109915339807072082741172...

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

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.

Cf. A274412 (V_1), A274413 (V_2A), A274414 (V_2N), A274415 (V_3T), A274416 (V_3S), A274417 (V_3L), A274419 (V_4N), A274420 (V_5), A274421 (V_6).

C0 = A143298 = (9 - PolyGamma[1, 2/3] + PolyGamma[1, 4/3])/(4*Sqrt[3]);
V4A = 6 Zeta[3] - 77/12 Zeta[4] - 6 C0^2;
RealDigits[V4A, 10, 100][[1]]

V_4A = 6 zeta(3) - 77/12 zeta(4) - 6 C^2, where C is A143298.

A274419 Decimal expansion of V_4N, a Quantum Field Theory constant [negated] related to the coloring of the tetrahedron with four masses (non-adjacent case).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 21 2016

			-6.054167858590219739369399569161448794813115362003582403652113814498...

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

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.

Cf. A274412 (V_1), A274413 (V_2A), A274414 (V_2N), A274415 (V_3T), A274416 (V_3S), A274417 (V_3L), A274418 (V_4A), A274420 (V_5), A274421 (V_6).

U = A255685 = Pi^4/180 + (Pi^2/12)*Log[2]^2 - (1/12)*Log[2]^4 - 2*PolyLog[4, 1/2];
V4N = 6 Zeta[3] - 14 Zeta[4] - 16 U;
RealDigits[V4N, 10, 103][[1]]

V_4N = 6*zeta(3) - 14*zeta(4) - 16*U, where U is A255685.

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

Original entry on oeis.org

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

Jean-François Alcover, Jun 21 2016

			-8.21685981750873806291339833860108582496950833917257503683557579115...

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

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.

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).

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]]

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

A274421 Decimal expansion of V_6, a Quantum Field Theory constant [negated] related to the coloring of the tetrahedron with six masses.

Original entry on oeis.org

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

Offset: 2

Jean-François Alcover, Jun 21 2016

			-10.03527847976878917191470068515890023865033349600275134054452580...

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

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.

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), A274420 (V_5).

C0 = A143298 = (9 - PolyGamma[1, 2/3] + PolyGamma[1, 4/3])/(4*Sqrt[3]);
U = A255685 = Pi^4/180 + (Pi^2/12)*Log[2]^2 - (1/12)*Log[2]^4 - 2*PolyLog[4, 1/2];
V6 = 6 Zeta[3] - 13 Zeta[4] - 8 U - 4 C0^2;
RealDigits[V6, 10, 101][[1]]

V6 = 6 zeta(3) - 13 zeta(4) - 8 U - 4 C^2, where U is A255685 and C A143298.

cp's OEIS Frontend

A274412 Decimal expansion of V_1, a Quantum Field Theory constant related to the coloring of the tetrahedron with one mass.

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A274413 Decimal expansion of V_2A, a Quantum Field Theory constant related to the coloring of the tetrahedron with two masses (adjacent case).

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A274414 Decimal expansion of V_2N, a Quantum Field Theory constant related to the coloring of the tetrahedron with two masses (non-adjacent case).

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A274415 Decimal expansion of V_3T, a Quantum Field Theory constant [negated] related to the coloring of the tetrahedron with three masses (triangle case).

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A274417 Decimal expansion of V_3L, a Quantum Field Theory constant [negated] related to the coloring of the tetrahedron with three masses (line case).

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A274418 Decimal expansion of V_4A, a Quantum Field Theory constant [negated] related to the coloring of the tetrahedron with four masses (adjacent case).

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A274419 Decimal expansion of V_4N, a Quantum Field Theory constant [negated] related to the coloring of the tetrahedron with four masses (non-adjacent case).

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