OFFSET
1,2
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.
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)
EXAMPLE
1.547982402157742304656076767753020632552256776913612065251441160613289...
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 *)
PROG
(PARI) default(realprecision, 100); 2*Pi*Catalan - 7*zeta(3)/2 \\ G. C. Greubel, Aug 24 2018
(Magma) SetDefaultRealField(RealField(100)); R:=RealField(); L:=RiemannZeta(); 2*Pi(R)*Catalan(R) - 7*Evaluate(L, 3)/2; // G. C. Greubel, Aug 24 2018
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Jul 11 2014
STATUS
approved