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

Author

Rick L. Shepherd, Jul 02 2002

Keywords

• nonn
• cons

Examples

			0.822467033424113218236207583323..
		

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.
• Eugène-Charles Catalan, Mémoire sur la transformation des séries et sur quelques intégrales définies, Mémoires de l'Académie royale de Belgique, 1867, Vol. 33, pp. 1-50.
• 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

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. Inverse of A377363.

Programs

Mathematica

RealDigits[Pi^2/12, 10, 105][[1]] (* Robert G. Wilson v *)

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

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 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 = A013661/2.

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

Equals 1 + (1/8)*Sum_{k >= 0} (-1)^(k-1) * (10*k + 13)/((k + 1)*(2*k + 1)^2*(2*k + 3)^2*binomial(2*k, k)). See Catalan, Section 35, equation 54. - Peter Bala, Aug 17 2024

Equals Integral_{x=0..oo} ((arctan(x) - Pi/4)*log(x^2 + 1))/(x^2) dx. - Kritsada Moomuang, Jun 04 2025