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%I #32 Oct 16 2021 06:12:57
%S 1,7,7,2,6,3,7,2,0,4,8,2,6,6,5,2,1,5,3,0,3,1,2,5,0,5,5,1,1,5,7,8,5,8,
%T 4,8,1,3,4,3,3,8,6,0,4,5,3,7,2,2,4,6,0,5,3,8,3,1,5,9,0,5,1,0,8,7,9,9,
%U 6,8,6,8,0,8,3,9,6,3,4,0,1,2,5,4,0,3,3,8,7,1,7,4,2,4,9,6,0,0,2,9,6,4,0,5,1,9,0,7,1,3,4,7,3,5,1
%N Decimal expansion of Sum_{n = -oo..oo} exp(-n^2).
%C A Riemann sum approximation to Integral_{-oo..oo} exp(-x^2) dx = sqrt(Pi).
%D Mentioned by N. D. Elkies in a lecture on the Poisson summation formula in Nashville TN in May 2010.
%H G. C. Greubel, <a href="/A195907/b195907.txt">Table of n, a(n) for n = 1..5000</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DedekindEtaFunction.html">Dedekind Eta Function</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F Equals Jacobi theta_{3}(0,exp(-1)). - _Jianing Song_, Oct 13 2021
%F Equals eta(i/Pi)^5 / (eta(i/(2*Pi))*eta(2*i/Pi))^2, where eta(t) = 1 - q - q^2 + q^5 + q^7 - q^12 - q^15 + ... is the Dedekind eta function without the q^(1/24) factor in powers of q = exp(2*Pi*i*t) (Cf. A000122). - _Jianing Song_, Oct 14 2021
%e 1.77263720482665215303125055115785848134338604537224605383159051...
%e For comparison, sqrt(Pi) = 1.7724538509055160272981674833411451827975494561223871282138... (A002161).
%t N[Sum[Exp[-n^2], {n, -Infinity, Infinity}], 200]
%t RealDigits[ N[ EllipticTheta[3, 0, 1/E], 115]][[1]] (* _Jean-François Alcover_, Nov 08 2012 *)
%o (PARI) 1 + 2*suminf(n=1,exp(-n^2)) \\ _Charles R Greathouse IV_, Jun 06 2016
%o (PARI) (eta(I/Pi))^5 / (eta(I/(2*Pi))^2 * eta(2*I/Pi)^2) \\ _Jianing Song_, Oct 13 2021
%Y Cf. A002161, A218792.
%K nonn,cons
%O 1,2
%A _N. J. A. Sloane_, Sep 25 2011
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