A102753 Decimal expansion of (Pi^2)/2.

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

Offset

1,1

Comments

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

Pi^2/2 is the squared side-length of a square with diagonal Pi. - Wesley Ivan Hurt, Jan 28 2022

References

J. Rivaud, Analyse, Séries, Equations différentielles, Mathématiques Supérieures et Spéciales, Premier Cycle Universitaire, Vuibert, 1981, Exercice 2, p. 135.

David 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 and 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.

Index entries for transcendental numbers

Formula

Equals psi_1(1/2), where psi_1(x) is the second logarithmic derivative of GAMMA(x).

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].

From Peter Bala, Nov 05 2019: (Start)

Pi^2/2 = Integral_{x = 0..inf} cosh(x)*x^2/sinh(x)^2 dx.

Pi^2/2 = 5*sum_{k >= 0} binomial(2*k,k)(-1/16)^k*1/(2*k+1)^2.

Pi^2/2 = 10*Integral_{x = 0..1/2} 1/x*log(x + sqrt(1 + x^2)) dx. (End)

Pi^2/20 = 0.1 * Pi^2/2 = Sum_{k>=1} 1/A026424(k)^2. - Amiram Eldar, Aug 17 2020

Conjecture: Pi^2/2 = Sum_{n = -oo..oo} ( cos(Pi*sqrt(n^2+1)) - cos(Pi*n) ) (using the Eisenstein summation convention). - Peter Bala, Oct 08 2021

Pi^2/2 = Integral_{x = -oo..oo} x/sinh(x) dx (see Rivaud reference). - Bernard Schott, Jan 28 2022

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, A000796, A019699, A026424, A164103, A164105, A164106, A164108, A248359, A276023.

Sequence in context: A296448 A217316 A159628 * A200416 A199178 A318331

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