cp's OEIS Frontend

A319332 Decimal expansion of 1/2 + Sum_{n>0} exp(-Pi*n^2).

%I A319332 #23 Aug 06 2024 06:11:17
%S A319332 5,4,3,2,1,7,4,0,5,6,0,6,6,5,4,0,0,7,2,8,7,6,5,8,0,6,0,7,5,5,1,1,1,7,
%T A319332 2,8,5,3,5,1,0,2,8,5,3,6,2,2,6,0,9,4,4,2,9,6,0,3,9,5,1,5,7,9,9,0,9,2,
%U A319332 8,3,6,6,1,3,3,5,5,4,8,9,7,9,8,0,2,8,0,8
%N A319332 Decimal expansion of 1/2 + Sum_{n>0} exp(-Pi*n^2).
%C A319332 A part of Ramanujan's question 629 in the Journal of the Indian Mathematical Society (VII, 40) asked "... deduce the following: 1/2 + Sum_{n>=1} exp(-Pi*n^2) = sqrt(5*sqrt(5)-10) * (1/2 + Sum_{n>=1} exp(-5*Pi*n^2))."
%H A319332 B. C. Berndt, Y. S. Choi, and S. Y. Kang, <a href="https://faculty.math.illinois.edu/~berndt/jims.ps">The problems submitted by Ramanujan to the Journal of Indian Math. Soc.</a>, in: Continued fractions, Contemporary Math., 236 (1999), 15-56, DOI: 10.1090/conm/236 (see Q629, JIMS VII).
%H A319332 B. C. Berndt, Y. S. Choi, and S. Y. Kang, <a href="https://citeseerx.ist.psu.edu/pdf/ae75da0be9fb455e2c55daa5fca46ae63e6a60bd">The problems submitted by Ramanujan to the Journal of Indian Math. Soc.</a>, in: Continued fractions, Contemporary Math., 236 (1999), 15-56 (see Q629, JIMS VII).
%H A319332 Dan Romik, <a href="https://arxiv.org/abs/1807.06130">The Taylor coefficients of the Jacobi theta_3</a>, arXiv:1807.06130 [math.NT], 2018.
%F A319332 Equals Pi^(1/4)/(2*Gamma(3/4)). - _Peter Luschny_, Jun 11 2020
%F A319332 From _Amiram Eldar_, May 30 2023: (Start)
%F A319332 Equals Gamma(1/4)/(2*sqrt(2)*Pi^(3/4)).
%F A319332 Equals A327996 / sqrt(Pi). (End)
%e A319332 0.54321740560665400728765806075511172853510285362260944296039515799...
%t A319332 RealDigits[Pi^(1/4)/(2*Gamma[3/4]), 10, 120][[1]] (* _Amiram Eldar_, May 30 2023 *)
%o A319332 (PARI) 1/2+suminf(n=1,exp(-Pi*n*n))
%o A319332 (PARI) sqrt(5*sqrt(5)-10)*(1/2+suminf(n=1,exp(-5*Pi*n*n)))
%Y A319332 Cf. A175573, A327996.
%K A319332 nonn,cons
%O A319332 0,1
%A A319332 _Hugo Pfoertner_, Sep 18 2018