A273084 Decimal expansion of theta_3(0, exp(-6*Pi)), where theta_3 is the 3rd Jacobi theta function.

1, 0, 0, 0, 0, 0, 0, 0, 1, 3, 0, 2, 4, 8, 2, 4, 2, 7, 2, 1, 5, 9, 8, 0, 1, 4, 5, 6, 4, 2, 4, 3, 3, 0, 2, 3, 0, 9, 0, 6, 7, 4, 5, 7, 3, 2, 5, 4, 1, 4, 6, 0, 4, 1, 5, 7, 5, 1, 1, 4, 8, 0, 1, 1, 9, 0, 4, 5, 9, 3, 4, 8, 2, 3, 9, 1, 1, 1, 3, 6, 1, 2, 5, 1, 7, 1, 1, 8, 6, 0, 8, 8, 8, 1, 0, 9, 2, 6, 4, 0, 4, 4, 6, 7, 4

Offset

1,10

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Wikipedia, Theta function

Formula

Equals sqrt(2+sqrt(8+6*sqrt(3)+4*sqrt(6+4*sqrt(3)))) * Pi^(1/4) / (2*3^(3/8)*Gamma(3/4)).

Equals sqrt((A273081^2 + A292888^4/A363018^2)/2). - Vaclav Kotesovec, May 17 2023

Example

1.0000000130248242721598014564243302309067457325414604157511...

Maple

evalf(sqrt(2 + sqrt(8 + 6*sqrt(3) + 4*sqrt(6 + 4*sqrt(3)))) * Pi^(1/4) / (2*3^(3/8) * GAMMA(3/4)), 120);

Mathematica

RealDigits[EllipticTheta[3, 0, Exp[-6*Pi]], 10, 105][[1]]
RealDigits[Sqrt[2 + Sqrt[8 + 6*Sqrt[3] + 4*Sqrt[6 + 4*Sqrt[3]]]] * Pi^(1/4) / (2*3^(3/8) * Gamma[3/4]), 10, 105][[1]]

Prog

(PARI) th3(x)=1 + 2*suminf(n=1, x^n^2) th3(exp(-6*Pi)) \\ Charles R Greathouse IV, Jun 06 2016
(Magma) C<i> := ComplexField(); Sqrt(2+Sqrt(8+6*Sqrt(3)+4*Sqrt(6 +4*Sqrt(3))))*Pi(C)^(1/4)/(2*3^(3/8)*Gamma(3/4)) // G. C. Greubel, Jan 07 2018

Crossrefs

Cf. A175573, A247217, A273081, A273082, A273083, A273086.

Sequence in context: A092154 A177344 A139585 * A349728 A261163 A292244

Adjacent sequences: A273081 A273082 A273083 * A273085 A273086 A273087

Keyword

nonn,cons

Author

Vaclav Kotesovec, May 14 2016

Status

approved