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A319332
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Decimal expansion of 1/2 + Sum_{n>0} exp(-Pi*n^2).
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1
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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, 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, 8, 3, 6, 6, 1, 3, 3, 5, 5, 4, 8, 9, 7, 9, 8, 0, 2, 8, 0, 8
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OFFSET
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0,1
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COMMENTS
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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))."
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LINKS
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FORMULA
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Equals Gamma(1/4)/(2*sqrt(2)*Pi^(3/4)).
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EXAMPLE
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0.54321740560665400728765806075511172853510285362260944296039515799...
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MATHEMATICA
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RealDigits[Pi^(1/4)/(2*Gamma[3/4]), 10, 120][[1]] (* Amiram Eldar, May 30 2023 *)
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PROG
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(PARI) 1/2+suminf(n=1, exp(-Pi*n*n))
(PARI) sqrt(5*sqrt(5)-10)*(1/2+suminf(n=1, exp(-5*Pi*n*n)))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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