This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A247040 #16 Feb 16 2025 08:33:23 %S A247040 1,9,6,5,5,5,7,0,3,9,0,0,9,0,7,8,2,8,1,3,1,2,3,1,3,5,5,5,7,3,5,1,8,5, %T A247040 3,6,7,8,6,8,9,7,6,7,2,8,4,4,6,4,6,4,5,1,1,7,0,8,5,6,5,2,8,8,7,8,1,7, %U A247040 9,6,4,0,1,4,3,2,5,3,5,4,5,7,6,4,9,3,1,3,4,2,6,6,6,3,6,7,2,6,7,6,4,2,9,8 %N A247040 Decimal expansion of M_6, the 6th Madelung constant. %D A247040 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 77. %H A247040 G. C. Greubel, <a href="/A247040/b247040.txt">Table of n, a(n) for n = 1..5000</a> %H A247040 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/DirichletBetaFunction.html"> Dirichlet Beta Function</a> %H A247040 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/MadelungConstants.html"> Madelung Constants</a> %F A247040 M6 = (3/Pi^2)*(4*(sqrt(2)-1)*zeta(1/2)*beta(5/2) - (4*sqrt(2)-1)*zeta(5/2)*beta(1/2)), where beta is Dirichlet's "beta" function. %e A247040 -1.9655570390090782813123135557351853678689767284464645117... %t A247040 beta[x_] := (Zeta[x, 1/4] - Zeta[x, 3/4])/4^x; M6 = (3/Pi^2)*(4*(Sqrt[2]-1)*Zeta[1/2]*beta[5/2] - (4*Sqrt[2]-1)*Zeta[5/2]*beta[1/2]); RealDigits[M6, 10, 104][[1]] %o A247040 (PARI) th4(x)=1+2*sumalt(n=1, (-1)^n*x^n^2) %o A247040 intnum(x=0, [oo, 1], (th4(exp(-x))^6-1)/sqrt(Pi*x)) \\ _Charles R Greathouse IV_, Jun 07 2016 %o A247040 (PARI) th4(x)=1+2*sumalt(n=1, (-1)^n*x^n^2) %o A247040 intnum(x=0, [oo, 1], (th4(exp(-x))^6-1)/sqrt(Pi*x)) \\ _Charles R Greathouse IV_, Jun 06 2016 %Y A247040 Cf. A059750, A088537, A085469, A090734. %K A247040 nonn,cons,easy %O A247040 1,2 %A A247040 _Jean-François Alcover_, Sep 10 2014