A100074 Decimal expansion of Pi^2/(12*e^3).
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, 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, 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
Offset: 0
Examples
0.040948222423400563521941804633807242093727299745689...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..10000
- R. William Gosper, Mourad E. H. Ismail and Ruiming Zhang, On some strange summation formulas, Illinois J. Math., Vol. 37, No. 2 (1993), pp. 240-277.
- Jonathan Sondow and Eric Weisstein, e, MathWorld.
- Eric Weisstein's World of Mathematics, Series.
Programs
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Mathematica
Join[{0}, RealDigits[Pi^2*Exp[-3]/12, 10, 120][[1]]] (* Amiram Eldar, Jun 09 2021 *) -
SageMath
numerical_approx(pi^2*exp(-3)/12, digits=120) # G. C. Greubel, Jun 08 2022
Formula
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
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