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%I #123 Oct 06 2023 08:34:44
%S 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,
%T 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,
%U 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
%N Decimal expansion of Pi^2/12.
%D C. C. Clawson, The Beauty and Magic of Numbers. New York: Plenum Press (1996): 98
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.11 p. 126 and section 8.5 p. 501.
%D Jolley, Summation of Series, Dover (1961) eq. (234) page 44.
%H Ivan Panchenko, <a href="/A072691/b072691.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&Page=807&Submit=Go">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Paul Bracken, <a href="https://cms.math.ca/publications/crux/issue/?volume=49&issue=3">Problem 4826</a>, Crux Mathematicorum, Vol. 49, No. 3 (March, 2023), p. 157; Michel Bataille, <a href="https://cms.math.ca/publications/crux/issue/?volume=49&issue=8">Solution to Problem 4826</a>, ibid., Vol. 49, No. 8 (Oct. 2023), p. 452.
%H Brian Hawthorn, <a href="https://www.youtube.com/watch?v=gjjo2uKfRvs">The Hardest Integral I've Ever Done</a>, YouTube video, 2021.
%H Michael Penn, <a href="https://www.youtube.com/watch?v=x0d-tGtztbs">A viewer suggested integral</a>, YouTube video, 2021.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Dilogarithm.html">Dilogarithm</a>
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Equals 1/(1*2) + 1/(2*4) + 1/(3*6) + 1/(4*8) + ... [Jolley]
%F Equals -dilogarithm(-1). - _Rick L. Shepherd_, Jul 21 2004
%F Equals zeta(1,1), the double zeta-function with both arguments equal to 1. - _R. J. Mathar_, Oct 10 2011
%F Equals Sum_{n>=1} ((-1)^(n+1))/n^2 [Clawson]. - _Alonso del Arte_, Aug 15 2012
%F Equals Integral_{x=0..1} log((1+x^3)/(1-x^3))/x dx. - _Bruno Berselli_, May 13 2013
%F From _Jean-François Alcover_, May 17 2013: (Start)
%F Equals zeta(2)/2.
%F Equals Integral_{x=1..2} log(x)/(x-1) dx. (End)
%F Equals lim_{n->infinity} A244583(n)/prime(n)^2. See A244583 for details. - _Richard R. Forberg_, Jan 04 2015
%F 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
%F 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
%F Equals lim_{n->infinity} A024916(n)/(n^2). - _Omar E. Pol_, Dec 15 2021
%F Integral_{x=0..1} -log(x)/(x+1) dx. - _Bernard Schott_, Apr 25 2022
%F 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
%e 0.822467033424113218236207583323... = A013661/2.
%t RealDigits[Pi^2/12, 10, 105][[1]] (* _Robert G. Wilson v_ *)
%o (PARI) zeta(2)/2 \\ _Michel Marcus_, Sep 08 2014
%o (PARI) -dilog(-1) \\ _Charles R Greathouse IV_, Apr 17 2015
%o (PARI) Pi^2/12 \\ _Charles R Greathouse IV_, Apr 17 2015
%o (PARI) sumnumrat(1/(2*x^2), 0) \\ _Charles R Greathouse IV_, Jan 20 2022
%o (Python)
%o from mpmath import *
%o mp.dps=106
%o print([int(c) for c in list(str(zeta(2)/2))[2:-1]]) # _Indranil Ghosh_, Jul 08 2017
%Y Cf. A072692 (Pi^2/12 is in asymptotic formula related to sigma(n), A000203).
%Y Cf. A113319 (sum_{i>=0} 1/(i^2+1)); A232883 (sum_{i>=0} 1/(2*i^2+1)).
%Y Cf. A001008, A002805, A013661, A024916, A237593, A244583.
%K nonn,cons
%O 0,1
%A _Rick L. Shepherd_, Jul 02 2002
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