cp's OEIS Frontend

A157697 Decimal expansion of sqrt(2/3).

%I A157697 #74 Jun 10 2025 01:44:04
%S A157697 8,1,6,4,9,6,5,8,0,9,2,7,7,2,6,0,3,2,7,3,2,4,2,8,0,2,4,9,0,1,9,6,3,7,
%T A157697 9,7,3,2,1,9,8,2,4,9,3,5,5,2,2,2,3,3,7,6,1,4,4,2,3,0,8,5,5,7,5,0,3,2,
%U A157697 0,1,2,5,8,1,9,1,0,5,0,0,8,8,4,6,6,1,9,8,1,1,0,3,4,8,8,0,0,7,8,2,7,2,8,6,4
%N A157697 Decimal expansion of sqrt(2/3).
%C A157697 Height (from a vertex to the opposite face) of regular tetrahedron with unit edge. - _Stanislav Sykora_, May 31 2012
%C A157697 The eccentricity of the ellipse of minimum area that is circumscribing two equal and externally tangent circles (Kotani, 1995). - _Amiram Eldar_, Mar 06 2022
%C A157697 The standard deviation of a roll of a 3-sided die. - _Mohammed Yaseen_, Feb 23 2023
%D A157697 L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (168) on page 32.
%H A157697 G. C. Greubel, <a href="/A157697/b157697.txt">Table of n, a(n) for n = 0..10000</a>
%H A157697 Jisho Kotani, <a href="https://cms.math.ca/publications/crux/issue?volume=21&amp;issue=6">Problem 2053</a>, Crux Mathematicorum, Vol. 5, No. 1 (1995), p. 202; <a href="https://cms.math.ca/publications/crux/issue?volume=22&amp;issue=4">Solution to Problem 2053</a>, by David Hankin, ibid., Vol. 22, No. 4 (1996), pp. 187-188.
%H A157697 D. H. Lehmer, <a href="http://www.jstor.org/stable/2322496">Interesting series involving the Central Binomial Coefficient</a>, Am. Math. Monthly, Vol. 92, No. 7 (1985), pp. 449-457.
%H A157697 <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>.
%F A157697 Equals 1 - (1/2)/2 + (1*3)/(2*4)/2^2 - (1*3*5)/(2*4*6)/2^3 + ... [Jolley]
%F A157697 Equals Sum_{n>=0} (-1)^n*binomial(2n,n)/8^n = 1/A115754. Averaging this constant with sqrt(2) = A002193 = Sum_{n>=0} binomial(2n,n)/8^n yields A145439.
%F A157697 From _Michal Paulovic_, Dec 08 2022: (Start)
%F A157697 Equals 2 * A020763.
%F A157697 Has periodic continued fraction expansion [0, 1, 4; 2, 4]. (End)
%F A157697 Equals exp(-arctanh(1/5)). - _Amiram Eldar_, Jul 10 2023
%F A157697 Equals Product_{k>=1} (1 + (-1)^k/A092259(k)). - _Amiram Eldar_, Nov 24 2024
%e A157697 0.81649658092772603273242802490196379732198249355222...
%p A157697 evalf(sqrt(2/3)) ;
%t A157697 RealDigits[Sqrt[2/3], 10, 200][[1]] (* _Vladimir Joseph Stephan Orlovsky_, Mar 04 2011*)
%o A157697 (PARI) sqrt(2/3) \\ _G. C. Greubel_, Mar 30 2018
%o A157697 (Magma) Sqrt(2/3); // _G. C. Greubel_, Mar 30 2018
%Y A157697 Cf. A002193, A020763, A092259, A115754, A145439.
%K A157697 cons,easy,nonn
%O A157697 0,1
%A A157697 _R. J. Mathar_, Mar 04 2009