A218792 Decimal expansion of Sum_{n = -oo..oo} e^(-2*n^2).
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, 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, 7, 7, 5, 8, 5, 6, 6, 5, 5, 3, 0, 7, 5, 0, 6, 2, 3, 9, 3
Offset: 1
Examples
1.2713415221890152252223825787909356249768649877176... For comparison, sqrt(Pi/2) = 1.2533141373155002512078826424055226265034933703050...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..5000
- Eric Weisstein's World of Mathematics, Dedekind Eta Function
- Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Programs
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Mathematica
RealDigits[Sum[E^(-2*k^2), {k,-Infinity,Infinity}], 10, 200][[1]] RealDigits[EllipticTheta[3,0,1/E^2],10,200][[1]] (* Vaclav Kotesovec, Sep 22 2013 *) -
PARI
1 + 2*suminf(n=1, exp(-2*n^2)) \\ Charles R Greathouse IV, Jun 06 2016
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PARI
(eta(2*I/Pi))^5 / (eta(I/Pi)^2 * eta(4*I/Pi)^2) \\ Jianing Song, Oct 13 2021
Formula
Equals Jacobi theta_{3}(0,exp(-2)). - G. C. Greubel, Feb 01 2017
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