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

1,2

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(-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

Example

1.2713415221890152252223825787909356249768649877176...
For comparison, sqrt(Pi/2) = 1.2533141373155002512078826424055226265034933703050...

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 *)

Prog

(PARI) 1 + 2*suminf(n=1, exp(-2*n^2)) \\ Charles R Greathouse IV, Jun 06 2016
(PARI) (eta(2*I/Pi))^5 / (eta(I/Pi)^2 * eta(4*I/Pi)^2) \\ Jianing Song, Oct 13 2021

Crossrefs

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

Sequence in context: A201889 A145057 A344930 * A065254 A010592 A096381

Adjacent sequences: A218789 A218790 A218791 * A218793 A218794 A218795

Keyword

nonn,cons

Author

Vaclav Kotesovec, Nov 05 2012

Status

approved