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A157697 Decimal expansion of sqrt(2/3). 9

%I #66 Aug 21 2023 12:27:32

%S 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 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 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 Decimal expansion of sqrt(2/3).

%C Height (from a vertex to the opposite face) of regular tetrahedron with unit edge. - _Stanislav Sykora_, May 31 2012

%C 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 The standard deviation of a roll of a 3-sided die. - _Mohammed Yaseen_, Feb 23 2023

%D L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (168) on page 32.

%H G. C. Greubel, <a href="/A157697/b157697.txt">Table of n, a(n) for n = 0..10000</a>

%H 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 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 <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>

%F Equals 1 - (1/2)/2 + (1*3)/(2*4)/2^2 - (1*3*5)/(2*4*6)/2^3 + ... [Jolley]

%F 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 From _Michal Paulovic_, Dec 08 2022: (Start)

%F Equals 2 * A020763.

%F Has periodic continued fraction expansion [0, 1, 4; 2, 4]. (End)

%F Equals exp(-arctanh(1/5)). - _Amiram Eldar_, Jul 10 2023

%e 0.8164965809277260327324280249...

%p evalf(sqrt(2/3)) ;

%t RealDigits[Sqrt[2/3], 10, 200]][[1]] (* _Vladimir Joseph Stephan Orlovsky_, Mar 04 2011*)

%o (PARI) sqrt(2/3) \\ _G. C. Greubel_, Mar 30 2018

%o (Magma) Sqrt(2/3); // _G. C. Greubel_, Mar 30 2018

%Y Cf. A002193, A020763, A115754, A145439.

%K cons,easy,nonn

%O 0,1

%A _R. J. Mathar_, Mar 04 2009

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Last modified August 9 07:48 EDT 2024. Contains 375027 sequences. (Running on oeis4.)