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%I #51 May 13 2024 09:16:00
%S 5,4,8,3,1,1,3,5,5,6,1,6,0,7,5,4,7,8,8,2,4,1,3,8,3,8,8,8,8,2,0,0,8,3,
%T 9,6,4,0,6,3,1,6,6,3,3,7,3,5,5,9,9,4,7,9,2,4,5,1,8,6,0,7,6,4,5,6,6,6,
%U 9,1,5,6,8,0,1,0,6,6,9,5,7,9,4,4,5,4,2,9,6,6,8,7,3,2,5,2,9,0,1,7,6,8
%N Decimal expansion of Pi^2/18.
%C The sequence of repeating coefficients [1,-1,-2,-1,1,2] in the sum in the formula section, is equal to the 6th column in A191898. - _Mats Granvik_, Mar 19 2012
%D A. Holroyd, Sharp Metastability Threshold for Two-Dimensional Bootstrap Percolation, Prob. Th. and Related Fields 125, 195-224, 2003.
%H J. M. Borwein, R. Girgensohn, <a href="http://dx.doi.org/10.1007/s00010-005-2774-x">Evaluations of binomial series</a>, Aequat. Math. 70 (2005) 25-36
%H A. Holroyd, <a href="http://arxiv.org/abs/math/0206132">Sharp Metastability Threshold for Two-Dimensional Bootstrap Percolation</a>, arXiv:math/0206132 [math.PR], 2002.
%H Ji-Cai Liu, <a href="https://arxiv.org/abs/2002.03650">On two congruences involving Franel numbers</a>, arXiv:2002.03650 [math.NT], 2020.
%H Courtney Moen, <a href="http://www.jstor.org/stable/2690456">Infinite series with binomial coefficients</a>, Math. Mag. 64 (1) (1991) 53-55.
%H Renzo Sprugnoli, <a href="https://www.emis.de/journals/INTEGERS/papers/g27/g27.Abstract.html">Sums of reciprocals of the central binomial coefficients</a>, El. J. Combin. Numb. Th. 6 (2006) # A27
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CentralBinomialCoefficient.html">Central Binomial Coefficient</a>
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/BootstrapPercolation.html">Bootstrap Percolation</a>
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Sum[1/n^2/Binomial[2n,n], {n,Infinity}].
%F Pi^2/18 = A013661/3 = Sum[1/(i+0)^2 - 1/(i+1)^2 - 2/(i+2)^2 - 1/(i+3)^2 + 1/(i+4)^2 + 2/(i+5)^2, {i =1, 7, 13, 19, 25,.. infinity, stride of 6}]. - _Mats Granvik_, Mar 19 2012
%F Equals Sum_{k>=1} (H(k) - 2*H(2k))/((-3^k)*k). See Liu. - _Michel Marcus_, Feb 11 2020
%F Equals Sum_{k>=1} A007814(k)/k^2. - _Amiram Eldar_, Jul 13 2020
%F Equals (2/9) * Sum_{k>=0} (-1)^k*(7*k+5)*k!^3/((2*k+1)*(3*k+2)!) [Gosper 1974] - _R. J. Mathar_, Feb 07 2024
%F Continued fraction expansion: 1/(2 - 2/(13 - 48/(34 - 270/(65 - ... - 2*(2*n - 1)*n^3/(5*n^2 + 6*n + 2 - ... ))))). See A130549. - _Peter Bala_, Feb 16 2024
%e 0.54831...
%t RealDigits[Pi^2/18,10,120][[1]] (* _Harvey P. Dale_, Aug 14 2011 *)
%o (PARI) Pi^2/18 \\ _Charles R Greathouse IV_, Mar 20 2012
%Y Cf. A007814, A073010, A073016, A086464, A112093, A112094, A130549, A130550.
%K nonn,easy,cons,changed
%O 0,1
%A _Eric W. Weisstein_, Jul 21 2003
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