A276484 Decimal expansion of Sum_{k>=0} (2*k+2)/binomial(4*k+2, 2*k+1).

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

Offset: 1

Author

Ilya Gutkovskiy, Sep 05 2016

Keywords

• nonn
• cons

Examples

			1.22636467121674271390992581093973546...
		

Links

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

Crossrefs

Cf. A000108, A001622, A024492, A121839, A268813.

Programs

Magma

SetDefaultRealField(RealField(100)); R:=RealField(); 2*Pi(R)/(9*Sqrt(3)) + 6*(2*Sqrt(5)*Log((1+Sqrt(5))/2) + 15)/125; // G. C. Greubel, Nov 04 2018

Mathematica

RealDigits[2 (Pi/(9 Sqrt[3])) + 6 ((2 Sqrt[5] Log[GoldenRatio] + 15)/125), 10, 120][[1]]
RealDigits[HypergeometricPFQ[{1, 3/2, 2}, {3/4, 5/4}, 1/16], 10, 120][[1]]

PARI

suminf(k=0,(2*k+2)/binomial(4*k+2,2*k+1)) \\ Indranil Ghosh, Mar 04 2017

PARI

default(realprecision, 100); 2*Pi/(9*sqrt(3)) + 6*(2*sqrt(5)*log((1+sqrt(5))/2) + 15)/125 \\ G. C. Greubel, Nov 04 2018

Formula

Equals 2*Pi/(9*sqrt(3)) + 6*(2*sqrt(5)*log(phi) + 15)/125, where phi is the golden ratio (A001622).

Equals Sum_{k>=0} 1/Catalan number(2k+1).

Equals Sum_{k>=0} 1/A000108(2k+1).

Equals Sum_{k>=0} 1/A024492(k).