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%I #13 Sep 08 2022 08:46:17
%S 1,5,7,9,7,6,8,3,7,9,5,5,4,0,2,0,7,7,5,2,4,2,9,9,7,8,5,9,1,2,3,4,4,4,
%T 8,6,0,6,2,7,8,9,5,5,3,5,7,6,6,4,9,5,0,5,5,2,0,7,1,8,1,8,5,4,0,1,6,9,
%U 2,3,7,9,2,9,8,4,0,7,3,6,3,6,7,5,8,6,0,3,4,4,4,9,6,4,2,3,6,1,3,7,1,1,4,9,7,4,5,3,9,6,1,6,7,0,3,2,1,3,2,7
%N Decimal expansion of Sum_{k>=0} (2*k+1)/binomial(4*k,2*k).
%H G. C. Greubel, <a href="/A276483/b276483.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)) - 4*(3*sqrt(5)*log(phi) - 40)/125, where phi is the golden ratio (A001622).
%F Equals Sum_{k>=0} 1/Catalan number(2k).
%F Equals Sum_{k>=0} 1/A000108(2k).
%F Equals Sum_{k>=0} 1/A048990(k).
%e 1.57976837955402077524299785912344486...
%t RealDigits[2 (Pi/(9 Sqrt[3])) - 4 ((3 Sqrt[5] Log[GoldenRatio] - 40)/125), 10, 120][[1]]
%t RealDigits[HypergeometricPFQ[{1, 1, 3/2}, {1/4, 3/4}, 1/16], 10, 120][[1]]
%o (PARI) suminf(k=0, 1/(binomial(4*k,2*k)/(2*k+1))) \\ _Michel Marcus_, Sep 06 2016
%o (PARI) default(realprecision, 100); 2*Pi/(9*sqrt(3)) - 4*(3*sqrt(5)*log((1+sqrt(5))/2) - 40)/125 \\ _G. C. Greubel_, Nov 04 2018
%o (Magma) SetDefaultRealField(RealField(100)); R:=RealField(); 2*Pi(R)/(9*Sqrt(3)) - 4*(3*Sqrt(5)*Log((1+Sqrt(5))/2) - 40)/125; // _G. C. Greubel_, Nov 04 2018
%Y Cf. A000108, A001622, A048990, A121839, A268813.
%K nonn,cons
%O 1,2
%A _Ilya Gutkovskiy_, Sep 05 2016
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