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

%I A273084 #16 May 17 2023 04:33:32
%S A273084 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,
%T A273084 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,
%U A273084 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
%N A273084 Decimal expansion of theta_3(0, exp(-6*Pi)), where theta_3 is the 3rd Jacobi theta function.
%H A273084 G. C. Greubel, <a href="/A273084/b273084.txt">Table of n, a(n) for n = 1..10000</a>
%H A273084 Wikipedia, <a href="http://en.wikipedia.org/wiki/Theta_function#Explicit_values">Theta function</a>
%F A273084 Equals sqrt(2+sqrt(8+6*sqrt(3)+4*sqrt(6+4*sqrt(3)))) * Pi^(1/4) / (2*3^(3/8)*Gamma(3/4)).
%F A273084 Equals sqrt((A273081^2 + A292888^4/A363018^2)/2). - _Vaclav Kotesovec_, May 17 2023
%e A273084 1.0000000130248242721598014564243302309067457325414604157511...
%p A273084 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);
%t A273084 RealDigits[EllipticTheta[3, 0, Exp[-6*Pi]], 10, 105][[1]]
%t A273084 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]]
%o A273084 (PARI) th3(x)=1 + 2*suminf(n=1,x^n^2) th3(exp(-6*Pi)) \\ _Charles R Greathouse IV_, Jun 06 2016
%o A273084 (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
%Y A273084 Cf. A175573, A247217, A273081, A273082, A273083, A273086.
%K A273084 nonn,cons
%O A273084 1,10
%A A273084 _Vaclav Kotesovec_, May 14 2016