OFFSET
1,2
COMMENTS
A Riemann sum approximation to Integral_{-oo..oo} exp(-x^2) dx = sqrt(Pi).
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
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
EXAMPLE
1.77263720482665215303125055115785848134338604537224605383159051...
For comparison, sqrt(Pi) = 1.7724538509055160272981674833411451827975494561223871282138... (A002161).
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 *)
PROG
(PARI) 1 + 2*suminf(n=1, exp(-n^2)) \\ Charles R Greathouse IV, Jun 06 2016
(PARI) (eta(I/Pi))^5 / (eta(I/(2*Pi))^2 * eta(2*I/Pi)^2) \\ Jianing Song, Oct 13 2021
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
N. J. A. Sloane, Sep 25 2011
STATUS
approved