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%I #20 Sep 08 2022 08:46:17
%S 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,
%T 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,
%U 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
%N Decimal expansion of Sum_{k>=0} (2*k+2)/binomial(4*k+2, 2*k+1).
%H G. C. Greubel, <a href="/A276484/b276484.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/CatalanNumber.html">Catalan Number</a>
%F Equals 2*Pi/(9*sqrt(3)) + 6*(2*sqrt(5)*log(phi) + 15)/125, where phi is the golden ratio (A001622).
%F Equals Sum_{k>=0} 1/Catalan number(2k+1).
%F Equals Sum_{k>=0} 1/A000108(2k+1).
%F Equals Sum_{k>=0} 1/A024492(k).
%e 1.22636467121674271390992581093973546...
%t RealDigits[2 (Pi/(9 Sqrt[3])) + 6 ((2 Sqrt[5] Log[GoldenRatio] + 15)/125), 10, 120][[1]]
%t RealDigits[HypergeometricPFQ[{1, 3/2, 2}, {3/4, 5/4}, 1/16], 10, 120][[1]]
%o (PARI) suminf(k=0,(2*k+2)/binomial(4*k+2,2*k+1)) \\ _Indranil Ghosh_, Mar 04 2017
%o (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
%o (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
%Y Cf. A000108, A001622, A024492, A121839, A268813.
%K nonn,cons
%O 1,2
%A _Ilya Gutkovskiy_, Sep 05 2016
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