login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A111003 Decimal expansion of Pi^2/8. 11
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 (list; constant; graph; refs; listen; history; text; internal format)
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

REFERENCES

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.

C. C. Clawson, The Beauty and Magic of Numbers. New York: Plenum Press (1996): 98

L. B. W. Jolley, Summation of Series, Dover (1961).

LINKS

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

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]. - 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..infinity} 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

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

CROSSREFS

Sequence in context: A103356 A028257 A100228 * A289277 A140182 A239472

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 20 22:25 EST 2018. Contains 299387 sequences. (Running on oeis4.)