cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Jean-François Alcover, Jun 21 2016

Keywords

Examples

			-8.21685981750873806291339833860108582496950833917257503683557579115...

References

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

Links

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).

Programs

  • 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]]

Formula

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

A274441 Decimal expansion of Q(3), value of one of five integrals related to Quantum Field Theory (see the paper by David Broadhurst).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 23 2016

Keywords

Examples

			2.03436897131720444715410048232706998197695047365128645708443723938...

Links

Crossrefs

Cf. A274438 (Q(0)), A274439 (Q(1)), A274440 (Q(2)), A274442 (Q(4)).

Programs

  • Mathematica
    digits = 101;
Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]);
U = A255685 = Pi^4/180 + (Pi^2/12)*Log[2]^2 - (1/12)*Log[2]^4 - 2*PolyLog[4, 1/2];
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"];
Q[3] = -50/9  Cl2[Pi/3]^2 + 596/81 Zeta[4] - 16/9 U + 32/3 V;
RealDigits[N[Q[3], digits] // Chop][[1]]
  • PARI
    Q(n) = intnum(x=0, oo, acosh((x+2)/2)^2 * log((x+1)/x)/(x+n));
Q(3) \\ Gheorghe Coserea, Sep 30 2018
  • PARI
    clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));
polygamma(n, x) = if (n == 0, psi(x), (-1)^(n+1)*n!*zetahurwitz(n+1, x));
u31=Pi^4/180 + (Pi^2/12)*log(2)^2  - (1/12)*log(2)^4 - 2*polylog(4, 1/2);
v31=3*zeta(3)/8 - 1/2 + sumalt(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)));
-50/9*clausen(2, Pi/3)^2 + 596/81*zeta(4) - 16/9*u31 + 32/3*v31 \\ Gheorghe Coserea, Sep 30 2018

Formula

Q(n) = Integral_{0..inf} arccosh((x+2)/2)^2 log((x+1)/x)/(x+n) dx.
Computation is done using the analytical form given by David Broadhurst:
Q(3) = -50/9 Cl2(Pi/3)^2+596/81 zeta(4)-16/9 U+32/3 V, where Cl_2 is the Clausen integral, U A255685 and V A274400.

A274442 Decimal expansion of Q(4), value of one of five integrals related to Quantum Field Theory (see the paper by David Broadhurst).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 23 2016

Keywords

Examples

			1.87447166949008260118095099948968029705739765892037953480769845119...

Links

Crossrefs

Cf. A274438 (Q(0)), A274439 (Q(1)), A274440 (Q(2)), A274441 (Q(3)).

Programs

  • Mathematica
    digits = 101;
U = A255685 = Pi^4/180 + (Pi^2/12)*Log[2]^2 - (1/12)*Log[2]^4 - 2*PolyLog[4, 1/2];
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"];
Q[4] = 125/54 Zeta[4] + 8 U - 8 V;
RealDigits[Q[4], 10, digits][[1]]
  • PARI
    Q(n) = intnum(x=0, oo, acosh((x+2)/2)^2 * log((x+1)/x)/(x+n));
Q(4) \\ Gheorghe Coserea, Sep 30 2018
  • PARI
    polygamma(n, x) = if (n == 0, psi(x), (-1)^(n+1)*n!*zetahurwitz(n+1, x));
u31=Pi^4/180 + (Pi^2/12)*log(2)^2  - (1/12)*log(2)^4 - 2*polylog(4, 1/2);
v31=3*zeta(3)/8 - 1/2 + sumalt(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)));
125/54*zeta(4) + 8*u31 - 8*v31 \\ Gheorghe Coserea, Sep 30 2018

Formula

Q(n) = Integral_{0..inf} arccosh((x+2)/2)^2 log((x+1)/x)/(x+n) dx.
Computation is done using the analytical form given by David Broadhurst:
Q(4) = 125/54 zeta(4) + 8 U - 8 V, where U is A255685 and V A274400.
Showing 1-3 of 3 results.

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