A195907 Decimal expansion of Sum_{n = -oo..oo} exp(-n^2).
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, 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, 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
Offset: 1
Examples
1.77263720482665215303125055115785848134338604537224605383159051... For comparison, sqrt(Pi) = 1.7724538509055160272981674833411451827975494561223871282138... (A002161).
References
- Mentioned by N. D. Elkies in a lecture on the Poisson summation formula in Nashville TN in May 2010.
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
N[Sum[Exp[-n^2], {n, -Infinity, Infinity}], 200] RealDigits[ N[ EllipticTheta[3, 0, 1/E], 115]][[1]] (* Jean-François Alcover, Nov 08 2012 *) -
PARI
1 + 2*suminf(n=1,exp(-n^2)) \\ Charles R Greathouse IV, Jun 06 2016
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PARI
(eta(I/Pi))^5 / (eta(I/(2*Pi))^2 * eta(2*I/Pi)^2) \\ Jianing Song, Oct 13 2021
Formula
Equals Jacobi theta_{3}(0,exp(-1)). - Jianing Song, Oct 13 2021
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
Equals Product_{k>=1} tanh((k*(1 + i*Pi))/2), i=sqrt(-1). - Antonio Graciá Llorente, May 13 2024
Comments