A247037 Decimal expansion of Sum_{k >= 0} 1/(4*k+3)^2.
1, 5, 8, 8, 6, 7, 4, 7, 7, 9, 7, 9, 4, 7, 5, 4, 0, 6, 1, 4, 9, 8, 5, 3, 9, 3, 0, 0, 2, 6, 0, 6, 7, 3, 9, 0, 5, 7, 0, 0, 3, 1, 5, 2, 5, 8, 1, 1, 7, 1, 3, 3, 4, 7, 0, 1, 7, 5, 8, 5, 2, 7, 6, 2, 0, 2, 8, 7, 1, 2, 9, 1, 5, 1, 3, 0, 7, 2, 9, 4, 2, 9, 4, 7, 9, 3, 2, 5, 8, 1, 2, 6, 9, 3, 5, 1, 9, 6, 1, 3, 6, 4, 7
Offset: 0
Examples
0.158867477979475406149853930026067390570031525811713347...
References
- Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.7 Catalan's constant p. 55.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..10000
Programs
-
Magma
SetDefaultRealField(RealField(100)); R:= RealField(); (Pi(R)^2 - 8*Catalan(R))/16; // G. C. Greubel, Aug 24 2018
-
Mathematica
RealDigits[Pi^2/16 - Catalan/2 , 10, 103] // First
-
PARI
Pi^2/16 - Catalan/2 \\ Charles R Greathouse IV, Jan 30 2018
-
PARI
zetahurwitz(2,3/4)/16 \\ Charles R Greathouse IV, Jan 30 2018
-
PARI
sumpos(k=0,1/(4*k+3)^2) \\ Charles R Greathouse IV, Jan 30 2018
Formula
Equals Pi^2/16 - G/2, where G is Catalan's constant.
Equals zeta(2, 3/4)/16 = Psi(1, 3/4)/16, with the Hurwitz zeta function and the Trigamma function Psi(1, z), and the partial sums of the series given in the name are {A173955(n+2) / A173954(n+2)}{n>=0}. - _Wolfdieter Lang, Nov 14 2017
Equals Integral_{x=1..oo} log(x)/(x^4 - 1) dx. - Amiram Eldar, Jul 17 2020