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 1-5 of 5 results.

A284867 Decimal expansion of Ai(0), where Ai is the Airy function of the first kind.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 04 2017

Keywords

Examples

			0.35502805388781723926006318600418317639797917419917724058332651030081...
		

References

  • Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 56, page 559.

Crossrefs

Cf. A096714, A096715, A269892, A269893, A073006 (Gamma(2/3)), A284868 (Ai'(0)).

Programs

Formula

Ai(0) = 1/(3^(2/3)*Gamma(2/3)).

A358559 Decimal expansion of Bi(0), where Bi is the Airy function of the second kind.

Original entry on oeis.org

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

Views

Author

Dumitru Damian, Nov 22 2022

Keywords

Examples

			0.61492662744600073515092236909361355359472818864859650504087875301429651...
		

References

  • F. W. J. Olver, Asymptotics and Special Functions, Academic Press, ISBN 978-0-12-525856-2, 1974.
  • Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 56, page 559.

Crossrefs

Cf. A284867 (Ai(0)), A284868 (Ai'(0)), this sequence (Bi(0)), A358561 (Bi'(0)), A358564(Gi(0)).

Programs

  • Mathematica
    RealDigits[AiryBi[0], 10, 120][[1]] (* Amiram Eldar, Nov 28 2022 *)
  • PARI
    airy(0)[2]
    
  • PARI
    airy(0)[1]*sqrt(3)
    
  • PARI
    3^(1/3)*gamma(1/3)/(2*Pi)
    
  • SageMath
    airy_bi(0).n(algorithm='scipy', prec=250)

Formula

Bi(0) = A284867*A002194.
Bi(0) = A358564*3.
Bi(0) = 1/(3^(1/6)*A073006).
Bi(0) = A073005/(3^(1/6)*A186706).
Bi(0) = A073005/(3^(1/6)*2*A093602).
Bi(0) = 3^(1/3)*A073005/(2*A000796).
Bi(0) = A252799/(3^(1/6)*BarnesG[5/3]).
Bi(0) = 3^(1/4)/(2^(2/9) * Pi^(1/3) * AGM(2,(sqrt(2+sqrt(3))))^(1/3)), where AGM is the arithmetic-geometric mean.

A358561 Decimal expansion of the derivative Bi'(0), where Bi is the Airy function of the second kind.

Original entry on oeis.org

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

Views

Author

Dumitru Damian, Nov 22 2022

Keywords

Examples

			0.44828835735382635791482371039882839086622679921226206108280877837233075...
		

References

  • F. W. J. Olver, Asymptotics and Special Functions, Academic Press, ISBN 978-0-12-525856-2, 1974.

Crossrefs

Cf. A284867 (Ai(0)), A284868 (Ai'(0)), A358559 (Bi(0)), this sequence (Bi'(0)), A358564 (Gi(0)).

Programs

  • Mathematica
    RealDigits[AiryBi'[0], 10, 120][[1]] (* Amiram Eldar, Nov 28 2022 *)
  • PARI
    derivnum(x=0, airy(x)[2])
    
  • SageMath
    airy_bi_prime(0).n(algorithm='scipy', prec=250)

Formula

Bi'(0) = A284868*A002194.
Bi'(0) = 3*Gi'(0), where Gi' is the derivative of the inhomogeneous Airy function of the first kind.
Bi'(0) = 3^(1/6)/A073005.
Bi'(0) = A073006*3^(1/6)/A186706.
Bi'(0) = A073006*3^(1/6)/2*A093602.
Bi'(0) = 3^(2/3)*A073006/(2*A000796).
Bi'(0) = 3^(1/4)*AGM(2,(sqrt(2+sqrt(3))))^(1/3)/(2^(7/9) * Pi^(2/3)), where AGM is the arithmetic-geometric mean.

A358564 Decimal expansion of Gi(0), where Gi is the inhomogeneous Airy function of the first kind (also called Scorer function).

Original entry on oeis.org

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

Views

Author

Dumitru Damian, Nov 22 2022

Keywords

Examples

			0.204975542482000245050307456364537851198242729549532168346959584338098839...
		

References

  • Scorer, R. S., Numerical evaluation of integrals of the form Integral_{x=x1..x2} f(x)*e^(i*phi(x))dx and the tabulation of the function Gi(z)=(1/Pi)*Integral_{u=0..oo} sin(u*z+u^3/3) du, Quart. J. Mech. Appl. Math. 3 (1950), 107-112.

Crossrefs

Cf. A284867 (Ai(0)), A284868 (Ai'(0)), A358559 (Bi(0)), A358561 (Bi'(0)), this sequence (Gi(0)).

Programs

  • Mathematica
    First[RealDigits[N[ScorerGi[0],90]]] (* Stefano Spezia, Nov 28 2022 *)
  • PARI
    airy(0)[2]/3
    
  • PARI
    1/(3^(7/6)*gamma(2/3))
    
  • PARI
    sqrt(3)*gamma(1/3)/(3^(7/6)*2*Pi)
    
  • PARI
    1/(3^(3/4)*2^(2/9)*Pi^(1/3)*sqrtn(agm(2,(sqrt(2+sqrt(3)))),3))
    
  • SageMath
    1/(3^(7/6)*gamma(2/3)).n(algorithm='scipy', prec=250)

Formula

Gi(0) = A358559/3.
Gi(0) = A284867/A002194.
Gi(0) = Hi(0)/2, where Hi is the inhomogeneous Airy function of the second kind.
Gi(0) = 1/(3^(7/6)*A073006).
Gi(0) = A073005/(3^(7/6)*A186706).
Gi(0) = A073005/(3^(7/6)*2*A093602).
Gi(0) = A073005/(3^(4/6)*2*A000796).
Gi(0) = A252799/(3^(7/6)*BarnesG(5/3)).
Gi(0) = 1/(3^(3/4) * 2^(2/9) * Pi^(1/3) * AGM(2,(sqrt(2+sqrt(3))))^(1/3)), where AGM is the arithmetic-geometric mean.

A385253 Decimal expansion of the function AiryAi(x) at x=1.

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Jul 29 2025

Keywords

Comments

Second derivative at x=1 has that same value.

Examples

			0.1352924163128814155241474235154663...
		

Crossrefs

Programs

  • Maple
    AiryAi(1) ; evalf(%) ; # R. J. Mathar, Jul 31 2025
  • Mathematica
    RealDigits[AiryAi[1], 10, 105][[1]]
  • PARI
    airy(1)[1] \\ Michel Marcus, Jul 29 2025

Formula

Equals AiryAi''(1).
Showing 1-5 of 5 results.