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A102753 Decimal expansion of (Pi^2)/2. 15
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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} 4^k/(k^2*binomial(2*k,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

Volume of the 4-dimensional unit sphere; the volume of the n-dimensional unit sphere is Pi^(n/2)/gamma(n/2+1) (see n-ball link and A164103). - Rick L. Shepherd, Jun 22 2017

REFERENCES

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Middlesex, England: Penguin Books, 1986, p. 53.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000

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, Hypersphere

Eric Weisstein's World of Mathematics, Trigamma Function

Wikipedia, Hypersphere.

Wikipedia, Volume of an n-ball.

EXAMPLE

4.9348022005446793094172454999380755676568497036203953132066746881100\ 224112096026215008867018592761159120129568870115720388....

MATHEMATICA

RealDigits[Pi^2/2, 10, 111][[1]] (* Robert G. Wilson v, Dec 15 2005 *)

PROG

(PARI) Pi^2/2 \\ Michel Marcus, Sep 04 2015

CROSSREFS

Cf. A002388, A248359, A000796, A019699, A164103, A164105, A164106, A164108, A276023.

Sequence in context: A196819 A217316 A159628 * A200416 A199178 A198828

Adjacent sequences:  A102750 A102751 A102752 * A102754 A102755 A102756

KEYWORD

cons,nonn

AUTHOR

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Feb 10 2005

STATUS

approved

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Last modified August 21 05:59 EDT 2017. Contains 290862 sequences.