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

1,7

Links

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

Wikipedia, Theta function

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

Example

1.00000697468471241799127935745572277338608481181934395967...

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]]

Prog

(PARI) (2^.25+1)*gamma(1/4)/sqrtn(128*Pi^3, 4) \\ Charles R Greathouse IV, Jun 06 2016
(Magma) C<i> := ComplexField(); ((2+2^(3/4))/4)*Pi(C)^(1/4)/Gamma(3/4) // G. C. Greubel, Jan 07 2018

Crossrefs

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

Sequence in context: A229921 A145429 A194789 * A065414 A198135 A019813

Adjacent sequences: A273079 A273080 A273081 * A273083 A273084 A273085

Keyword

nonn,cons

Author

Vaclav Kotesovec, May 14 2016

Status

approved