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.

User: Christian N. Hofmann

Christian N. Hofmann's wiki page.

Christian N. Hofmann has authored 1 sequences.

A363747 Decimal expansion of 2*Integral_{x=0..1} 1/sqrt(1-x^16) dx.

Original entry on oeis.org

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

Views

Author

Christian N. Hofmann, Jun 19 2023

Keywords

Comments

Let I(k) = 2*Integral_{x=0..1} 1/sqrt(1-x^(2^k)) dx. Then I(1) = Pi (cf. A000796), I(2) = Gauss lemniscate constant (cf. A062539), I(3) = sqrt(2)*K(sqrt(2)-1) (cf. A262427), I(4) = this constant.
Also I(k) = (Pi^(3/2)*Product_{j=2..k} cos(Pi/(2^j)))/(Gamma(1/2+1/(2^k))*Gamma(1-1/(2^k))). - Christian N. Hofmann, Aug 25 2023

Examples

			2.1682048381784119930017239089489332786586588673422...
Gamma(1/16) = 2^(45/16)*Pi^(1/16)*I(2)^(1/8)*I(3)^(1/4)*I(4)^(1/2) = 15.481281...

Crossrefs

Cf. A000796, A062539, A262427.

Programs

  • Maple
    evalf(2*int(1/sqrt(1-t^16),t=0..1), 120);
  • Mathematica
    RealDigits[Beta[1/16, 1/2]/8, 10, 120][[1]] (* Amiram Eldar, Jun 22 2023 *)
  • PARI
    2*intnum(x=0, 1, 1/sqrt(1-x^16)) \\ Michel Marcus, Jun 22 2023

Formula

Equals Beta(1/16,1/2)/8 = 2*sqrt(Pi)*Gamma(17/16)/Gamma(9/16).
Equals Pi^(3/2)/(8*sin(Pi/16)*Gamma(9/16)*Gamma(15/16)). - Christian N. Hofmann, Aug 28 2023
Gamma(1/2^n) = 2^((n-1)*(1-1/2^n)) * Product_{k=1..n} I(k)^(1/(2^(n-k+1))).

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.