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%I #28 Dec 17 2024 09:11:25
%S 0,8,4,0,6,9,5,0,8,7,2,7,6,5,5,9,9,6,4,6,1,4,8,9,5,0,2,4,7,9,0,3,5,5,
%T 1,1,9,3,7,5,7,2,7,9,6,4,6,8,0,1,1,9,6,1,8,4,2,9,7,2,7,2,4,6,0,0,1,3,
%U 5,9,7,9,0,7,0,1,6,7,7,2,0,6,2,4,8,7,4,7,5,9,8,3,1,8,9,0,6,3,6,0,9,8
%N Decimal expansion of 2/3 + zeta(1/2)/sqrt(2*Pi).
%D David H. Bailey, Jonathan M. Borwein, Neil J. Calkin, Roland Girgensohn, D. Russell Luke and Victor H. Moll, Experimental Mathematics in Action, Wellesley, MA: A K Peters, 2007, pp. 18 and 227.
%D Jonathan Borwein, David Bailey and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, Wellesley, MA: A K Peters, 2004, pp. 15-17.
%H G. C. Greubel, <a href="/A096616/b096616.txt">Table of n, a(n) for n = 0..10000</a>
%H Jonathan M. Borwein and Scott B. Lindstrom, <a href="http://www.ybook.co.jp/online2/oppafa/vol1/p361.html">Meetings with Lambert W and other special functions in optimization and analysis</a>, Pure and Applied Functional Analysis, Vol. 1, No. 3 (2016), pp. 361-396, <a href="https://www.carmamaths.org/resources/jon/WinOpt.pdf">alternative link</a>.
%H Donald E. Knuth, <a href="http://www.jstor.org/stable/2695746">Problem 10832</a>, The American Mathematical Monthly, Vol. 107, No. 9 (2000), p. 863, <a href="http://www.jstor.org/stable/2695574">A Stirling Series</a>, solution by Cecil C. Rousseau, ibid., Vol. 108, No. 9 (2001), pp. 877-878.
%H Allen Stenger, <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.124.2.116">Experimental Math for Math Monthly Problems</a>, The American Mathematical Monthly, Vol. 124, No. 2 (2017), pp. 116-131, <a href="https://www.allenstenger.com/uploads/1/4/1/8/14182140/expmathmathmonthlyfeb2017.pdf">alternative link</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KnuthsSeries.html">Knuth's Series</a>.
%F Equals Sum_{k>=1} (1/sqrt(2*Pi*k) - k^k/(k!*exp(k))). - _Amiram Eldar_, Oct 13 2020
%F Equals 2/3 - A134469. - _R. J. Mathar_, Dec 17 2024
%e 0.0840695087...
%t Flatten[{0, RealDigits[2/3 + Zeta[1/2]/Sqrt[2*Pi], 10, 100][[1]]}] (* _Vaclav Kotesovec_, Aug 16 2015 *)
%o (PARI) 2/3 + zeta(1/2)/sqrt(2*Pi) \\ _Michel Marcus_, Aug 15 2015
%Y Cf. A019727, A059750, A231863.
%K nonn,cons,changed
%O 0,2
%A _Eric W. Weisstein_, Jun 30 2004