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%I #109 Nov 19 2023 10:33:54
%S 1,8,1,3,7,9,9,3,6,4,2,3,4,2,1,7,8,5,0,5,9,4,0,7,8,2,5,7,6,4,2,1,5,5,
%T 7,3,2,2,8,4,0,6,6,2,4,8,0,9,2,7,4,0,5,7,5,5,6,9,8,8,4,9,3,5,3,8,8,1,
%U 2,3,1,8,1,1,2,6,3,5,3,8,8,3,6,8,4,1,2,4,9,8,8,2,1,2,0,6,0,1,6,8,8,5,6,2,2
%N Decimal expansion of Pi/sqrt(3) = sqrt(2*zeta(2)).
%C Volume of a cube with edge length 1 rotated about a space diagonal. See MathWorld Cube page. - _Francis Wolinski_, Mar 10 2019
%C Volume of a cone with unit radius and 60-degree opening angle, and so height sqrt(3). Equivalently, the volume of the cone formed by rotating a 30-60-90 degree triangle with unit short leg about the long leg. - _Christoph B. Kassir_, Sep 17 2022
%H Harry J. Smith, <a href="/A093602/b093602.txt">Table of n, a(n) for n = 1..20000</a>
%H Peter Bala, <a href="/A002117/a002117.pdf">New series for old functions</a>
%H Jean Dolbeault, Ari Laptev and Michael Loss, <a href="http://doi.org/10.4171/JEMS/142">Lieb-Thirring inequalities with improved constants</a>, Vol. 10, No. 4 (2008), pp. 1121-1126, <a href="https://arxiv.org/abs/0708.1165">preprint</a>, arXiv:0708.1165 [math.AP], 2007.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/No-Three-in-a-Line-Problem.html">No-Three-in-a-Line-Problem</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Cube.html">Cube</a>
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Equals Integral_{x=0..oo} x^(1/3)/(1+x^2) dx. - _Jean-François Alcover_, May 24 2013
%F Equals (3/2)*Integral_{x=0..oo} 1/(1+x+x^2) dx. - _Bruno Berselli_, Jul 23 2013
%F Equals Sum_{n >= 0} (1/(6*n+1) - 4/(6*n+2) - 5/(6*n+3) - 1/(6*n+4) + 4/(6*n+5) + 5/(6*n+6)). - _Mats Granvik_, Sep 23 2013
%F Equals (1/2) * Sum_{n >= 0} (14*n + 11)*(-1/3)^n/((4*n + 1)*(4*n + 3)*binomial(4*n,2*n)). For more series representations of this type see the Bala link. - _Peter Bala_, Feb 04 2015
%F From _Peter Bala_, Nov 02 2019: (Start)
%F Equals 3*Sum_{n >= 1} 1/( (3*n - 1)*(3*n - 2) ).
%F Equals 2 - 6*Sum_{n >= 1} 1/( (3*n - 1)*(3*n + 1)*(3*n + 2) ).
%F Equals 5!*Sum_{n >= 1} 1/( (3*n - 1)*(3*n - 2)*(3*n + 2)*(3*n + 4) ).
%F Equals 3*( 1 - 2*Sum_{n >= 1} 1/(9*n^2 - 1) ).
%F Equals 1 + Sum_{n >=1 } (-1)^(n+1)*(6*n + 1)/(n*(n + 1)*(3*n + 1)*(3*n - 2)).
%F Equals (27/2)*Sum_{n >= 1} (2*n + 1)/( (3*n - 1)*(3*n + 1)*(3*n + 2)*(3*n + 4) ).
%F Equals 3*Integral_{x = 0..1} 1/(1 + x + x^2) dx.
%F Equals 3*Integral_{x = 0..1} (1 + x)/(1 - x + x^2) dx.
%F Equals 3*Integral_{x = 0..oo} cosh(x)/cosh(3*x) dx. (End)
%F Equals Integral_{x = 0..oo} log(1+x^3)/x^3 dx. - _Amiram Eldar_, Aug 20 2020
%F Equals (27*S - 36)/24, where S = A248682. - _Peter Luschny_, Jul 22 2022
%F From _Peter Bala_, Nov 09 2023: (Start)
%F For any integer k, Pi/sqrt(3) = Sum_{n >= 0} (1/(n + k + 1/3) - 1/(n - k + 2/3)) = (1/3)*Sum_{n >= 0} (1/(n - k + 1/6) - 1/(n + k + 5/6)).
%F Equals (3/2)*Sum_{n >= 0} 1/((2*n + 1)*binomial(2*n, n)). (End)
%e Pi/sqrt(3) = 1.8137993642342178505940782576421557322840662480927405755...
%t RealDigits[Pi/Sqrt[3],10,120][[1]] (* _Harvey P. Dale_, Mar 04 2012 *)
%o (PARI) default(realprecision, 20080); x=Pi*sqrt(3)/3; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b093602.txt", n, " ", d)); \\ _Harry J. Smith_, Jun 19 2009
%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)/Sqrt(3); // _G. C. Greubel_, Mar 10 2019
%o (Sage) numerical_approx(pi/sqrt(3), digits=100) # _G. C. Greubel_, Mar 10 2019
%Y Continued fraction expansion is A132116. - _Jonathan Vos Post_, Aug 10 2007
%Y Equals twice A093766.
%Y Cf. A343235 (using the reciprocal), A248682.
%K easy,nonn,cons
%O 1,2
%A _Lekraj Beedassy_, May 14 2004
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