A358519 - OEIS

%I #19 Aug 05 2024 14:21:34

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

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

%U 1,4,6,7,1,4,9,3,6,1,9,8,5,9,8,5,3,2,5,6,4,7,7,5,6,1,3

%N Decimal expansion of Sum_{k >= 1} (-1)^(k+1)/(k^2 + 4*k - 1).

%H Michael I. Shamos, <a href="https://citeseerx.ist.psu.edu/pdf/ae33a269baba5e8b1038e719fb3209e8a00abec5">A catalog of the real numbers</a> (2011), p. 253.

%F Equals Sum_{k >=1} (-1)^(k+1)/(k^2 + 4*k - 1).

%F Equals (Pi/(2*sqrt(5)))*csc(Pi*sqrt(5)) - 17/20 = Sum_{k>=3} (-1)^(k+1)/(k^2-5). [from the Shamos reference]

%F Equals Sum_{k>=3} (-1)^(k+1)/A028875(k). - _Amiram Eldar_, Nov 21 2022

%e 0.189957907718062725271908409063636655279566491...

%t RealDigits[(2*Sqrt[5]*Pi*Csc[Sqrt[5]*Pi] - 17)/20, 10, 120][[1]] (* _Amiram Eldar_, Nov 21 2022 *)

%o (PARI) (Pi/(2*sqrt(5)))*1/sin(Pi*sqrt(5)) - 17/20

%Y Cf. A028875, A242023, A242024, A358516, A358517.

%Y Cf. A002162.

%K nonn,cons

%O 0,2

%A _Claude H. R. Dequatre_, Nov 20 2022