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%I A020832 #49 Nov 22 2024 11:04:31
%S A020832 1,1,5,4,7,0,0,5,3,8,3,7,9,2,5,1,5,2,9,0,1,8,2,9,7,5,6,1,0,0,3,9,1,4,
%T A020832 9,1,1,2,9,5,2,0,3,5,0,2,5,4,0,2,5,3,7,5,2,0,3,7,2,0,4,6,5,2,9,6,7,9,
%U A020832 5,5,3,4,4,6,0,5,8,6,6,6,9,1,3,8,7,4,3,0,7,9,1,1,7,1,4,9,9,0,5
%N A020832 Decimal expansion of 1/sqrt(75).
%C A020832 Multiplied by 10 this is 2/sqrt(3). - _Alonso del Arte_, Apr 30 2012
%C A020832 2/sqrt(3) is Hermite's constant gamma_2. - _Jean-François Alcover_, Sep 02 2014, after Steven Finch.
%C A020832 2/sqrt(3) is the Lorentz factor for an object traveling at half the speed of light. - _Sean Stroud_, May 05 2019
%H A020832 Ivan Panchenko, <a href="/A020832/b020832.txt">Table of n, a(n) for n = 0..1000</a>
%H A020832 Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020, p. 62.
%H A020832 Yining Hu, <a href="http://arxiv.org/abs/1506.00151">Patterns in numbers and infinite sums and products</a>, arXiv:1506.00151 [math.NT], 2015.
%H A020832 Samuel G. Moreno and Esther M. García, <a href="http://www.jstor.org/stable/10.4169/math.mag.86.1.015">New Infinite Products of Cosines and Viète-Like Formulae</a>, Mathematics Magazine, vol. 86, no. 1, 2013, pp. 15-25. See formula for 2/sqrt(3) page 15.
%H A020832 <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>
%F A020832 (csc(Pi/3))/10, where csc is the cosecant function. - _Alonso del Arte_, Apr 30 2012
%F A020832 Product_{n>=1} ((3*n+1)/(3*n+2))^((-1)^n), with offset 1. (see Hu link). - _Michel Marcus_, Jun 02 2015
%F A020832 From _Amiram Eldar_, Aug 02 2020: (Start)
%F A020832 2/sqrt(3) = Sum_{k>=0} binomial(2*k,k)/16^k.
%F A020832 2/sqrt(3) = 1 + Sum_{k>=1} (2*k-1)!!/((2*k)!! * 2^(2*k)). (End)
%F A020832 2/sqrt(3) = Product_{k>=1} (1 - (-1)^k/A047235(k)). - _Amiram Eldar_, Nov 22 2024
%e A020832 0.1154700538379251529...
%t A020832 RealDigits[1/Sqrt[75], 10, 100][[1]] (* _Alonso del Arte_, Apr 30 2012 *)
%o A020832 (PARI) 75^-.5 \\ _Charles R Greathouse IV_, Mar 31 2016
%Y A020832 Cf. A010153 (continued fraction, but missing the initial 0), A047235.
%K A020832 nonn,cons,easy
%O A020832 0,3
%A A020832 _N. J. A. Sloane_.

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