A218792 - OEIS

%I #21 Oct 16 2021 05:24:03

%S 1,2,7,1,3,4,1,5,2,2,1,8,9,0,1,5,2,2,5,2,2,2,3,8,2,5,7,8,7,9,0,9,3,5,

%T 6,2,4,9,7,6,8,6,4,9,8,7,7,1,7,6,2,6,7,0,1,1,6,4,4,1,0,8,0,1,6,9,7,4,

%U 7,7,5,8,5,6,6,5,5,3,0,7,5,0,6,2,3,9,3

%N Decimal expansion of Sum_{n = -oo..oo} e^(-2*n^2).

%H G. C. Greubel, <a href="/A218792/b218792.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(-2)). - _G. C. Greubel_, Feb 01 2017

%F Equals eta(2*i/Pi)^5 / (eta(i/Pi)*eta(4*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 1.2713415221890152252223825787909356249768649877176...

%e For comparison, sqrt(Pi/2) = 1.2533141373155002512078826424055226265034933703050...

%t RealDigits[Sum[E^(-2*k^2), {k,-Infinity,Infinity}], 10, 200][[1]]

%t RealDigits[EllipticTheta[3,0,1/E^2],10,200][[1]] (* _Vaclav Kotesovec_, Sep 22 2013 *)

%o (PARI) 1 + 2*suminf(n=1, exp(-2*n^2)) \\ _Charles R Greathouse IV_, Jun 06 2016

%o (PARI) (eta(2*I/Pi))^5 / (eta(I/Pi)^2 * eta(4*I/Pi)^2) \\ _Jianing Song_, Oct 13 2021

%Y Cf. A195907, A069998, A167009, A167010, A229052.

%K nonn,cons

%O 1,2

%A _Vaclav Kotesovec_, Nov 05 2012