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%I #51 Feb 11 2024 22:37:13
%S 9,0,1,5,4,2,6,7,7,3,6,9,6,9,5,7,1,4,0,4,9,8,0,3,6,2,1,1,3,3,5,8,7,4,
%T 9,3,0,7,3,7,3,9,7,1,9,2,5,5,3,7,4,1,6,1,3,4,4,2,0,3,6,6,6,5,0,6,3,7,
%U 8,6,5,4,3,3,9
%N Decimal expansion of the Dirichlet eta-function at 3.
%C This constant is irrational by Apéry's theorem. - _Charles R Greathouse IV_, Feb 11 2024
%H Vincenzo Librandi, <a href="/A197070/b197070.txt">Table of n, a(n) for n = 0..10000</a>
%H Seán Stewart, <a href="https://doi.org/10.1080/00029890.2020.1792243">Problem 12206</a>, The American Mathematical Monthly, Vol. 127, No. 8 (2020), p. 752.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Dirichlet_eta_function">Dirichlet eta function</a>.
%F Equals 3*zeta(3)/4 = 3*A002117/4.
%F Also equals the integral over the unit cube [0,1]x[0,1]x[0,1] of 1/(1+x*y*z) dx dy dz. - _Jean-François Alcover_, Nov 24 2014
%F Equals Sum_{n>=1} (-1)^(n+1)/n^3. - _Terry D. Grant_, Aug 03 2016
%F Equals Lim_{n -> infinity} A136675(n)/A334582(n). - _Petros Hadjicostas_, May 07 2020
%F Equals Sum_{n>=1} AH(2*n)/n^2, where AH(n) = Sum_{k=1..n} (-1)^(k+1)/k = A058313(n)/A058312(n) is the n-th alternating harmonic number (Stewart, 2020). - _Amiram Eldar_, Oct 04 2021
%e 0.9015426773696957140498036211335874930737...
%p 3*Zeta(3)/4 ; evalf(%) ;
%t RealDigits[3(Zeta[3])/4, 10, 75][[1]] (* _Bruno Berselli_, Dec 20 2011 *)
%o (PARI) -polylog(3,-1) \\ _Charles R Greathouse IV_, Mar 28 2012
%o (PARI) 3/4*zeta(3) \\ _Charles R Greathouse IV_, Mar 28 2012
%Y Cf. A002117 (zeta(3)), A058312, A058313, A072691, A136675, A233090 (5*zeta(3)/8), A233091 (7*zeta(3)/8), A334582.
%K cons,easy,nonn
%O 0,1
%A _R. J. Mathar_, Oct 09 2011
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