A245073 Decimal expansion of Integral_{x=0..Pi/2} (x^2/sin(x)) dx.

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

Offset: 1

Author

Jean-François Alcover, Jul 11 2014

Keywords

• nonn
• cons
• easy

Examples

			1.547982402157742304656076767753020632552256776913612065251441160613289...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.7 Catalan's Constant, pp. 55, 57.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's MathWorld, Apery's Constant.
• Eric Weisstein's MathWorld, Catalan's Constant.

Crossrefs

Cf. A002117, A006752.

Programs

Magma

SetDefaultRealField(RealField(100)); R:=RealField(); L:=RiemannZeta();  2*Pi(R)*Catalan(R) - 7*Evaluate(L,3)/2; // G. C. Greubel, Aug 24 2018

Mathematica

RealDigits[2*Pi*Catalan - 7*Zeta[3]/2, 10, 105] // First
RealDigits[HypergeometricPFQ[{1,1,1,1},{3/2,3/2,2},1],10,100][[1]] (* Stefano Spezia, Nov 13 2024 *)

PARI

default(realprecision, 100); 2*Pi*Catalan - 7*zeta(3)/2 \\ G. C. Greubel, Aug 24 2018

Formula

Equals 2*Pi*G - 7*zeta(3)/2, where G is Catalan's constant.

Also equals 4 * Integral_{x=0..1} (arctan(x)^2/x) dx.

From Stefano Spezia, Nov 13 2024: (Start)

Equals Sum_{k>=0} 2^(4*k)/((k + 1)*(2*k + 1)^2*binomial(2*k,k)^2) (see Finch at p. 55).

Equals hypergeom([1, 1, 1, 1], [3/2, 3/2, 2], 1). (End)