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A195907
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Decimal expansion of Sum_{n = -oo..oo} exp(-n^2).
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4
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1, 7, 7, 2, 6, 3, 7, 2, 0, 4, 8, 2, 6, 6, 5, 2, 1, 5, 3, 0, 3, 1, 2, 5, 0, 5, 5, 1, 1, 5, 7, 8, 5, 8, 4, 8, 1, 3, 4, 3, 3, 8, 6, 0, 4, 5, 3, 7, 2, 2, 4, 6, 0, 5, 3, 8, 3, 1, 5, 9, 0, 5, 1, 0, 8, 7, 9, 9, 6, 8, 6, 8, 0, 8, 3, 9, 6, 3, 4, 0, 1, 2, 5, 4, 0, 3, 3, 8, 7, 1, 7, 4, 2, 4, 9, 6, 0, 0, 2, 9, 6, 4, 0, 5, 1, 9, 0, 7, 1, 3, 4, 7, 3, 5, 1
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OFFSET
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1,2
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COMMENTS
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A Riemann sum approximation to Integral_{-oo..oo} exp(-x^2) dx = sqrt(Pi).
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REFERENCES
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Mentioned by N. D. Elkies in a lecture on the Poisson summation formula in Nashville TN in May 2010.
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LINKS
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FORMULA
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Equals Jacobi theta_{3}(0,exp(-1)). - Jianing Song, Oct 13 2021
Equals eta(i/Pi)^5 / (eta(i/(2*Pi))*eta(2*i/Pi))^2, where eta(t) = 1 - q - q^2 + q^5 + q^7 - q^12 - q^15 + ... is the Dedekind eta function without the q^(1/24) factor in powers of q = exp(2*Pi*i*t) (Cf. A000122). - Jianing Song, Oct 14 2021
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EXAMPLE
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1.77263720482665215303125055115785848134338604537224605383159051...
For comparison, sqrt(Pi) = 1.7724538509055160272981674833411451827975494561223871282138... (A002161).
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MATHEMATICA
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N[Sum[Exp[-n^2], {n, -Infinity, Infinity}], 200]
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PROG
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(PARI) (eta(I/Pi))^5 / (eta(I/(2*Pi))^2 * eta(2*I/Pi)^2) \\ Jianing Song, Oct 13 2021
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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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