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A273086 Decimal expansion of theta_3(0, exp(-sqrt(6)*Pi)), where theta_3 is the 3rd Jacobi theta function. 5
1, 0, 0, 0, 9, 0, 9, 9, 2, 1, 8, 8, 7, 2, 5, 6, 7, 6, 2, 9, 1, 9, 2, 8, 6, 0, 0, 4, 1, 2, 1, 5, 6, 6, 6, 7, 1, 8, 0, 4, 5, 8, 8, 1, 4, 6, 7, 3, 0, 3, 0, 1, 3, 3, 0, 8, 5, 9, 2, 4, 1, 7, 9, 6, 8, 1, 3, 9, 5, 8, 5, 4, 2, 0, 8, 7, 9, 5, 0, 0, 5, 6, 3, 3, 2, 7, 5, 4, 2, 2, 0, 2, 2, 1, 8, 2, 9, 1, 1, 4, 7, 4, 2, 1, 8 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,5

LINKS

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

Eric Weisstein's MathWorld, Jacobi Theta Functions

Wikipedia, Theta function

FORMULA

Equals ((6 + sqrt(6*(3 + 2*sqrt(2)))) * Gamma(1/24) * Gamma(5/24) * Gamma(7/24) * Gamma(11/24))^(1/4) / (2*6^(3/8)*Pi^(3/4)).

Equals (4 - sqrt(2) + sqrt(6))^(1/4) * sqrt(Gamma(1/24)*Gamma(11/24)) / (2^(3/2)*3^(3/8)*Pi^(3/4)).

EXAMPLE

1.0009099218872567629192860041215666718045881467303013308592...

MAPLE

evalf(((6 + sqrt(6*(3 + 2*sqrt(2)))) * GAMMA(1/24) * GAMMA(5/24) * GAMMA(7/24) * GAMMA(11/24))^(1/4) / (2*6^(3/8)*Pi^(3/4)), 120);

evalf((4 - sqrt(2) + sqrt(6))^(1/4) * sqrt(GAMMA(1/24)*GAMMA(11/24)) / (2^(3/2)*3^(3/8)*Pi^(3/4)), 120);

MATHEMATICA

RealDigits[EllipticTheta[3, 0, Exp[-Sqrt[6]*Pi]], 10, 105][[1]]

RealDigits[((6 + Sqrt[6*(3 + 2*Sqrt[2])]) * Gamma[1/24] * Gamma[5/24] * Gamma[7/24] * Gamma[11/24])^(1/4) / (2*6^(3/8)*Pi^(3/4)), 10, 105][[1]]

RealDigits[(4 - Sqrt[2] + Sqrt[6])^(1/4) * Sqrt[Gamma[1/24]*Gamma[11/24]] / (2^(3/2)*3^(3/8)*Pi^(3/4)), 10, 105][[1]]

PROG

(PARI) th3(x)=1 + 2*suminf(n=1, x^n^2)

th3(exp(-sqrt(6)*Pi)) \\ Charles R Greathouse IV, Jun 06 2016

CROSSREFS

Cf. A175573, A247217, A273081, A273082, A273083, A273084, A086231, A273087.

Sequence in context: A176523 A242400 A197264 * A332093 A056965 A347688

Adjacent sequences:  A273083 A273084 A273085 * A273087 A273088 A273089

KEYWORD

nonn,cons

AUTHOR

Vaclav Kotesovec, May 14 2016

STATUS

approved

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Last modified September 28 10:48 EDT 2021. Contains 347714 sequences. (Running on oeis4.)