A244979 Decimal expansion of Pi/(2*sqrt(5)).

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

Offset: 0

Author

Jean-François Alcover, Jul 09 2014

Keywords

• nonn
• cons
• easy

Examples

			0.702481473104072639315637464320489479946650918706720241998972102619214188...
		

References

• George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), Chapter 13 A Master Formula, p. 250.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Beta Function
• Index entries for transcendental numbers

Crossrefs

Cf. A244976, A244977, A244978, A019692.

Programs

Mathematica

RealDigits[Pi/(2*Sqrt[5]), 10, 105] // First

PARI

Pi/sqrt(20) \\ Charles R Greathouse IV, Sep 30 2022

Formula

Equals Integral_(0..1) (1 + x^2)/(1 + 3*x^2 + x^4) dx.

From Peter Bala, Feb 16 2015: (Start)

Also equals beta(1/2, 1/2)/(2*sqrt(5)), where 'beta' is Euler's beta function.

Pi/(2*sqrt(5)) = Integral_{t = 0..a} (1 + t^2)*(1 + t^6)/(1 + t^10) dt = a + a^3/3 + a^7/7 + a^9/9 - a^11/11 - a^13/13 - a^17/17 - a^19/19 + ..., where a = 1/2(sqrt(5) - 1). Hint: differentiate atan( sqrt(5)*(t - t^3)/(1 - 3*t^2 + t^4) ). (End)

Equals (1/2)*Sum_{n >= 0} (-1)^n*( 1/(10*n + 1) + 1/(10*n + 3) + 1/(10*n + 7) + 1/(10*n + 9) ). Cf. A019692. - Peter Bala, Oct 30 2019

From Amiram Eldar, Aug 06 2020: (Start)

Equals Integral_{x=0..oo} 1/(x^2 + 5) dx.

Equals 0.1 * Integral_{x=0..oo} log(1 + 5/x^2) dx. (End)

Equals Integral_{x = 0..1} 2/(4*x^2 + 5*(1 - x)^2) dx. - Peter Bala, Jul 22 2022