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.

A244262 Decimal expansion of theta = 2.472548..., an auxiliary constant used to compute the best constant in Friedrichs' inequality in one dimension.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 24 2014

Keywords

Examples

			2.4725480752401227014376355093582...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.6 Sobolev Isoperimetric Constants, p. 223.

Links

Crossrefs

Cf. A244263.

Programs

  • Mathematica
    theta = t /. FindRoot[Cos[t] - t/(t^2 + 1)*Sin[t] == -1, {t, 2}, WorkingPrecision -> 101]; RealDigits[theta] // First

Formula

Theta is the unique solution of the equation cos(t) - t/(t^2 + 1)*sin(t) = -1, with 0 < t < Pi.

A246859 Decimal expansion of the best constant K for the integral inequality integral_{0..1} f(x)^2*f'(x)^2 dx <= K*integral_{0..1} f'(x)^4 dx.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Sep 05 2014

Keywords

Examples

			0.34611896560593345099609054206876591598395281385974864...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.6 Sobolev Isoperimetric Constants.

Links

Crossrefs

Cf. A242440, A244263, A244347, A244354, A245292.

Programs

  • Mathematica
    (* Using Boyd's formula *) I0[p_, q_, r_] := Integrate[(((q - 1)*t + 1)*t^(1/p - 1))/ (((r*(q - 1))*t)/(r - q) + 1)^((r*p + p + q)/(r*p)), {t, 0, 1}]; K[p_, q_, r_] := (beta = (((r - 1)*p + (r - q))/((r - 1)*(p + q)))^(1/r); (((r - q)*p^p)*beta^(p + q - r))/(I0[p, q, r]^p*((r - 1)*(p + q)))); RealDigits[K[2, 2, 4], 10, 104] // First

Formula

24/(2*sqrt(3) + 3*sqrt(2)*arcsinh(sqrt(2)))^2.

A248914 Decimal expansion of L = Integral_{t=0..1} 1/(1-2t^2/3) dt, an auxiliary constant associated with one of the integral inequalities studied by David Boyd.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Oct 16 2014

Keywords

Examples

			1.40382196515535516573036373889961027748003532830665702207...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.6 Sobolev Isoperimetric Constants.

Links

Crossrefs

Cf. A242440, A244263, A244347, A244354, A245292, A246859.

Programs

  • Mathematica
    RealDigits[Sqrt[3/2]*ArcTanh[Sqrt[2/3]], 10, 101] // First

Formula

L = sqrt(3/2)*arctanh(sqrt(2/3)).
K = A246859 = 2/(L+1)^2.
Showing 1-3 of 3 results.

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