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A102753
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Decimal expansion of (Pi^2)/2.
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13
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4, 9, 3, 4, 8, 0, 2, 2, 0, 0, 5, 4, 4, 6, 7, 9, 3, 0, 9, 4, 1, 7, 2, 4, 5, 4, 9, 9, 9, 3, 8, 0, 7, 5, 5, 6, 7, 6, 5, 6, 8, 4, 9, 7, 0, 3, 6, 2, 0, 3, 9, 5, 3, 1, 3, 2, 0, 6, 6, 7, 4, 6, 8, 8, 1, 1, 0, 0, 2, 2, 4, 1, 1, 2, 0, 9, 6, 0, 2, 6, 2, 1, 5, 0, 0, 8, 8, 6, 7, 0, 1, 8, 5, 9, 2, 7, 6, 1, 1, 5, 9, 1, 2, 0, 1
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OFFSET
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1,1
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COMMENTS
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Equals psi_1(1/2), where psi_1(x) is the second logarithmic derivative of GAMMA(x).
Also equals the volume of revolution of the sine or cosine curve for one half period, Integral_{0,Pi} Sin(x)^2 dx. - Robert G. Wilson v, Dec 15 2005
Equals sum_{k >=1} 2^(2k)/k^2/binomial(2k,k) [Amdeberhan]. - R. J. Mathar, Sep 28 2007
Equals 4*sum_{k >=1} 1/(2k-1)^2 [Wells].
Also equals the area under the peak-shaped even function f(x)=x/sinh(x).
Proof: For the upper half of the integral, write f(x) = 2x*exp(-x)/(1-exp(-2x)) = sum_{k=1..infinity} 2x*exp(-(2k-1)x) and integrate term by term from zero to infinity. - Stanislav Sykora, Nov 01 2013
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REFERENCES
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D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Middlesex, England: Penguin Books, 1986, p. 53.
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LINKS
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Table of n, a(n) for n=1..105.
T. Amdeberhan, L. Medina, V. H. Moll, The integrals in Gradshteyn and Ryzhik. Part 5: Some trigonometric integrals, equation 2.39, arXiv:0705.2379 [math.CA], 2007.
Eric Weisstein's World of Mathematics, Trigamma Function.
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EXAMPLE
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4.9348022005446793094172454999380755676568497036203953132066746881100\ 224112096026215008867018592761159120129568870115720388....
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MATHEMATICA
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RealDigits[Pi^2/2, 10, 111][[1]] (* Robert G. Wilson v, Dec 15 2005 *)
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PROG
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(PARI) Pi^2/2 \\ Michel Marcus, Sep 04 2015
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CROSSREFS
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Cf. A002388, A248359.
Sequence in context: A196819 A217316 A159628 * A200416 A199178 A198828
Adjacent sequences: A102750 A102751 A102752 * A102754 A102755 A102756
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KEYWORD
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cons,nonn
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AUTHOR
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Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Feb 10 2005
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STATUS
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approved
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