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

1,2

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

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

Table of n, a(n) for n=1..105.

Index entries for transcendental numbers

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A087989 A028257 A100228 * A347904 A289277 A140182

Adjacent sequences: A111000 A111001 A111002 * A111004 A111005 A111006

Keyword

cons,nonn

Author

Sam Alexander, Oct 01 2005

Extensions

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

Status

approved