A273082 Decimal expansion of theta_3(0, exp(-4*Pi)), where theta_3 is the 3rd Jacobi theta function.

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

Offset: 1

Author

Vaclav Kotesovec, May 14 2016

Keywords

• nonn
• cons

Examples

			1.00000697468471241799127935745572277338608481181934395967...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Wikipedia, Theta function

Crossrefs

Cf. A175573, A247217, A273081, A273083, A273084.

Programs

Magma

C := ComplexField(); ((2+2^(3/4))/4)*Pi(C)^(1/4)/Gamma(3/4) // G. C. Greubel, Jan 07 2018

Maple

evalf((2+2^(3/4))/4*Pi^(1/4)/GAMMA(3/4), 120);
evalf((1 + 2^(1/4)) * GAMMA(1/4) / (2^(7/4)*Pi^(3/4)), 120);

Mathematica

RealDigits[EllipticTheta[3, 0, Exp[-4*Pi]], 10, 105][[1]]
RealDigits[(2 + 2^(3/4))/4 * Pi^(1/4) / Gamma[3/4], 10, 105][[1]]
RealDigits[(1 + 2^(1/4)) * Gamma[1/4] / (2^(7/4)*Pi^(3/4)), 10, 105][[1]]

PARI

(2^.25+1)*gamma(1/4)/sqrtn(128*Pi^3,4) \\ Charles R Greathouse IV, Jun 06 2016

Formula

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

Equals (1 + 2^(1/4)) * Gamma(1/4) / (2^(7/4)*Pi^(3/4)).

Equals sqrt((A247217^2 + A259149^4/A259150^2)/2). - Vaclav Kotesovec, May 17 2023