login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A072691 Decimal expansion of Pi^2/12. 90
8, 2, 2, 4, 6, 7, 0, 3, 3, 4, 2, 4, 1, 1, 3, 2, 1, 8, 2, 3, 6, 2, 0, 7, 5, 8, 3, 3, 2, 3, 0, 1, 2, 5, 9, 4, 6, 0, 9, 4, 7, 4, 9, 5, 0, 6, 0, 3, 3, 9, 9, 2, 1, 8, 8, 6, 7, 7, 7, 9, 1, 1, 4, 6, 8, 5, 0, 0, 3, 7, 3, 5, 2, 0, 1, 6, 0, 0, 4, 3, 6, 9, 1, 6, 8, 1, 4, 4, 5, 0, 3, 0, 9, 8, 7, 9, 3, 5, 2, 6, 5, 2, 0, 0, 2 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
REFERENCES
C. C. Clawson, The Beauty and Magic of Numbers. New York: Plenum Press (1996): 98
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.11 p. 126 and section 8.5 p. 501.
Jolley, Summation of Series, Dover (1961) eq. (234) page 44.
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Paul Bracken, Problem 4826, Crux Mathematicorum, Vol. 49, No. 3 (March, 2023), p. 157; Michel Bataille, Solution to Problem 4826, ibid., Vol. 49, No. 8 (Oct. 2023), p. 452.
Brian Hawthorn, The Hardest Integral I've Ever Done, YouTube video, 2021.
Michael Penn, A viewer suggested integral, YouTube video, 2021.
Eric Weisstein's World of Mathematics, Dilogarithm
FORMULA
Equals 1/(1*2) + 1/(2*4) + 1/(3*6) + 1/(4*8) + ... [Jolley]
Equals -dilogarithm(-1). - Rick L. Shepherd, Jul 21 2004
Equals zeta(1,1), the double zeta-function with both arguments equal to 1. - R. J. Mathar, Oct 10 2011
Equals Sum_{n>=1} ((-1)^(n+1))/n^2 [Clawson]. - Alonso del Arte, Aug 15 2012
Equals Integral_{x=0..1} log((1+x^3)/(1-x^3))/x dx. - Bruno Berselli, May 13 2013
From Jean-François Alcover, May 17 2013: (Start)
Equals zeta(2)/2.
Equals Integral_{x=1..2} log(x)/(x-1) dx. (End)
Equals lim_{n->infinity} A244583(n)/prime(n)^2. See A244583 for details. - Richard R. Forberg, Jan 04 2015
Equals Sum_{k>=1} H(k)/(k*2^k), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Aug 20 2020
Equals Integral_{0..infinity} x/(exp(x) + 1) dx. See Abramowitz-Stegun, 23.2.8, for s=2, p. 801. - Wolfdieter Lang, Sep 16 2020
Equals lim_{n->infinity} A024916(n)/(n^2). - Omar E. Pol, Dec 15 2021
Integral_{x=0..1} -log(x)/(x+1) dx. - Bernard Schott, Apr 25 2022
Equals 1/2 + Sum_{k>=1} H(k)/(k*(k+1)*(k+2)), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number (Bracken, 2023). - Amiram Eldar, Oct 06 2023
Equals Integral_{x >= 0} x^2/cosh(x)^2 dx. - Peter Bala, Jun 20 2024
EXAMPLE
0.822467033424113218236207583323... = A013661/2.
MATHEMATICA
RealDigits[Pi^2/12, 10, 105][[1]] (* Robert G. Wilson v *)
PROG
(PARI) zeta(2)/2 \\ Michel Marcus, Sep 08 2014
(PARI) -dilog(-1) \\ Charles R Greathouse IV, Apr 17 2015
(PARI) Pi^2/12 \\ Charles R Greathouse IV, Apr 17 2015
(PARI) sumnumrat(1/(2*x^2), 0) \\ Charles R Greathouse IV, Jan 20 2022
(Python)
from mpmath import *
mp.dps=106
print([int(c) for c in list(str(zeta(2)/2))[2:-1]]) # Indranil Ghosh, Jul 08 2017
CROSSREFS
Cf. A072692 (Pi^2/12 is in asymptotic formula related to sigma(n), A000203).
Cf. A113319 (sum_{i>=0} 1/(i^2+1)); A232883 (sum_{i>=0} 1/(2*i^2+1)).
Sequence in context: A138997 A248498 A133918 * A021928 A185111 A319188
KEYWORD
nonn,cons,changed
AUTHOR
Rick L. Shepherd, Jul 02 2002
STATUS
approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 25 13:13 EDT 2024. Contains 373705 sequences. (Running on oeis4.)