A111003 Decimal expansion of Pi^2/8.

1, 2, 3, 3, 7, 0, 0, 5, 5, 0, 1, 3, 6, 1, 6, 9, 8, 2, 7, 3, 5, 4, 3, 1, 1, 3, 7, 4, 9, 8, 4, 5, 1, 8, 8, 9, 1, 9, 1, 4, 2, 1, 2, 4, 2, 5, 9, 0, 5, 0, 9, 8, 8, 2, 8, 3, 0, 1, 6, 6, 8, 6, 7, 2, 0, 2, 7, 5, 0, 5, 6, 0, 2, 8, 0, 2, 4, 0, 0, 6, 5, 5, 3, 7, 5, 2, 2, 1, 6, 7, 5, 4, 6, 4, 8, 1, 9, 0, 2, 8, 9, 7, 8, 0, 0

Offset: 1

Author

Sam Alexander, Oct 01 2005

Keywords

• cons
• nonn

Comments

According to Beckmann, Euler discovered the formula for this number as a sum of squares of reciprocals of odd numbers, along with similar formulas for Pi^2/6 and Pi^2/12. - Alonso del Arte, Apr 01 2013

Equals the asymptotic mean of the abundancy index of the odd numbers. - Amiram Eldar, May 12 2023

Examples

			1.23370055013616982735431137498451889191421242590509882830166867202...
1 + 1/9 + 1/25 + 1/49 + 1/81 + 1/121 + 1/169 + 1/225 + ... - _Bruno Berselli_, Mar 06 2017
		

References

• F. Aubonnet, D. Guinin and B. Joppin, Précis de Mathématiques, Analyse 2, Classes Préparatoires, Premier Cycle Universitaire, Bréal, 1990, Exercice 908, pages 82 and 91-92.
• Petr Beckmann, A History of Pi, 5th Ed. Boulder, Colorado: The Golem Press (1982): p. 153.
• George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 122.
• Calvin C. Clawson, The Beauty and Magic of Numbers. New York: Plenum Press (1996): 98.
• L. B. W. Jolley, Summation of Series, Dover (1961).
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 54.

Links

• Eric Weisstein's World of Mathematics, Dirichlet Lambda Function. See (5).
• Index entries for transcendental numbers

Crossrefs

Cf. A013661 (Pi^2/6), A002388, A102753, A091476, A309891.

Programs

Mathematica

RealDigits[Pi^2/8, 10, 105][[1]] (* Robert G. Wilson v *)

PARI

Pi^2/8 \\ Charles R Greathouse IV, Dec 04 2016

PARI

sumpos(n=1,(2*n-1)^-2) \\ Charles R Greathouse IV, Mar 02 2018

Formula

Equals 1 + 1/(2*3) + (1/3)*(1*2)/(3*5) + (1/4)*(1*2*3)/(3*5*7) + ... [Jolley eq 276]

Equals Sum_{k >= 1} 1/(2*k - 1)^2 [Clawson and Wells]. - Alonso del Arte, Aug 15 2012

Equals 2*(Integral_{t=0..1} sqrt(1 - t^2) dt)^2. - Alonso del Arte, Mar 29 2013

Equals Sum_{k >= 1} 2^k/(k^2*binomial(2*k, k)). - Jean-François Alcover, Apr 29 2013

Equals Integral_{x=0..1} log((1+x^2)/(1-x^2))/x dx. - Bruno Berselli, May 13 2013

Equals limit_{p->0} Integral_{x=0..Pi/2} x*tan(x)^p dx. [Jean-François Alcover, May 17 2013, after Boros & Moll p. 230]

Equals A002388/8 = A102753/4 = A091476/2. - R. J. Mathar, Oct 13 2015

Equals Integral_{x>=0} x*K_0(x)*K_1(x)dx where K are modified Bessel functions [Gradsteyn-Ryzhik 6.576.4]. - R. J. Mathar, Oct 22 2015

Equals (3/4)*zeta(2) = (3/4)*A013661. - Wolfdieter Lang, Sep 02 2019

From Amiram Eldar, Jul 17 2020: (Start)

Equals -Integral_{x=0..1} log(x)/(1 - x^2) dx = Integral_{x>=1} log(x)/(x^2-1) dx.

Equals -Integral_{x=0..oo} log(x)/(1 - x^4) dx.

Equals Integral_{x=0..oo} arctan(x)/(1 + x^2) dx. (End)

Equals Integral_{x=0..1} log(1+x+x^2+x^3)/x dx (Aubonnet). - Bernard Schott, Feb 04 2022

Equals Sum_{n>=1} A309891(n)/n^2. - Friedjof Tellkamp, Jan 25 2025

Equals lambda(2), where lambda is the Dirichlet lambda function. - Michel Marcus, Aug 15 2025

Extensions

More terms from Robert G. Wilson v, Oct 04 2005