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A100074 Decimal expansion of Pi^2/(12*e^3).

%I A100074 #29 Feb 16 2025 08:32:55
%S A100074 0,4,0,9,4,8,2,2,2,4,2,3,4,0,0,5,6,3,5,2,1,9,4,1,8,0,4,6,3,3,8,0,7,2,
%T A100074 4,2,0,9,3,7,2,7,2,9,9,7,4,5,6,8,9,6,1,8,4,7,7,7,8,1,7,0,0,3,0,2,3,0,
%U A100074 9,3,4,7,4,9,3,8,1,0,9,7,9,2,5,8,5,4,7,4,0,1,3,4,3,4,3,2,8,0,3,5,9,2,5
%N A100074 Decimal expansion of Pi^2/(12*e^3).
%H A100074 G. C. Greubel, <a href="/A100074/b100074.txt">Table of n, a(n) for n = 0..10000</a>
%H A100074 R. William Gosper, Mourad E. H. Ismail and Ruiming Zhang, <a href="http://doi.org/10.1215/ijm/1255987146">On some strange summation formulas</a>, Illinois J. Math., Vol. 37, No. 2 (1993), pp. 240-277.
%H A100074 Jonathan Sondow and Eric Weisstein, <a href="https://mathworld.wolfram.com/e.html">e</a>, MathWorld.
%H A100074 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Series.html">Series</a>.
%F A100074 Equals Sum_{k >= 1} (-1)^(k+1) * cos(sqrt(k^2*Pi^2 - 9))/k^2 (Gosper et al., 1993). - _Amiram Eldar_, Jun 09 2021
%F A100074 More generally, it appears that Pi^2/(12*exp(x)) = Sum_{k >= 1} (-1)^(k+1)*cos(sqrt(k^2*Pi^2*x/3 - x^2))/k^2 for 0 <= x <= 3. The above identity is the case x = 3. - _Peter Bala_, Jun 20 2022
%e A100074 0.040948222423400563521941804633807242093727299745689...
%t A100074 Join[{0}, RealDigits[Pi^2*Exp[-3]/12, 10, 120][[1]]] (* _Amiram Eldar_, Jun 09 2021 *)
%o A100074 (SageMath) numerical_approx(pi^2*exp(-3)/12, digits=120) # _G. C. Greubel_, Jun 08 2022
%Y A100074 Cf. A002388 (Pi^2), A091933 (e^3), A092035 (Pi^2/e^2).
%K A100074 nonn,cons
%O A100074 0,2
%A A100074 _Eric W. Weisstein_, Nov 02 2004