A072691 Decimal expansion of Pi^2/12.

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

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

Ivan Panchenko, Table of n, a(n) for n = 0..1000

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

Index entries for transcendental numbers

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

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)).

Cf. A001008, A002805, A013661, A024916, A237593, A244583.

Sequence in context: A138997 A248498 A133918 * A021928 A185111 A319188

Adjacent sequences: A072688 A072689 A072690 * A072692 A072693 A072694

Keyword

nonn,cons

Author

Rick L. Shepherd, Jul 02 2002

Status

approved