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A195907
Decimal expansion of Sum_{n = -oo..oo} exp(-n^2).
4
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
1,2
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
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
Sequence in context: A335929 A303658 A169812 * A126584 A318302 A266271
KEYWORD
nonn,cons
AUTHOR
N. J. A. Sloane, Sep 25 2011
STATUS
approved