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%I #49 Aug 14 2023 10:30:56
%S 5,7,7,3,5,0,2,6,9,1,8,9,6,2,5,7,6,4,5,0,9,1,4,8,7,8,0,5,0,1,9,5,7,4,
%T 5,5,6,4,7,6,0,1,7,5,1,2,7,0,1,2,6,8,7,6,0,1,8,6,0,2,3,2,6,4,8,3,9,7,
%U 7,6,7,2,3,0,2,9,3,3,3,4,5,6,9,3,7,1,5,3,9,5,5,8,5,7,4,9,5,2,5
%N Decimal expansion of 1/sqrt(3).
%C If the sides of a triangle form an arithmetic progression in the ratio 1:1+d:1+2d then when d=1/sqrt(3) it uniquely maximizes the area of the triangle. This triangle has approximate internal angles 25.588 degs, 42.941 degs, 111.471 degs. - _Frank M Jackson_, Jun 15 2011
%C When a cylinder is completely enclosed by a sphere, it occupies a fraction f of the sphere volume. The value of f has a trivial lower bound of 0, and an upper bound which is this constant. It is achieved iff the cylinder diameter is sqrt(2) times its height, and the sphere is circumscribed to it. A similar constant can be associated with any n-dimensional geometric shape. For 3D cuboids it is A165952. - _Stanislav Sykora_, Mar 07 2016
%C The ratio between the thickness and diameter of a dynamically fair coin having an equal probability, 1/3, of landing on each of its two faces and on its side after being tossed in the air. The calculation is based on the dynamic of rigid body (Yong and Mahadevan, 2011). See A020765 for a simplified geometrical solution. - _Amiram Eldar_, Sep 01 2020
%C The coefficient of variation (relative standard deviation) of natural numbers: Limit_{n->oo} sqrt((n-1)/(3*n+3)) = 1/sqrt(3). - _Michal Paulovic_, Mar 21 2023
%H Ivan Panchenko, <a href="/A020760/b020760.txt">Table of n, a(n) for n = 0..1000</a>
%H Ee Hou Yong and L. Mahadevan, <a href="https://doi.org/10.1119/1.3630934">Probability, geometry, and dynamics in the toss of a thick coin</a>, American Journal of Physics, Vol. 79, No. 12 (2011), pp. 1195-1201, <a href="https://arxiv.org/abs/1008.4559">preprint</a>, arXiv:1008.4559 [physics.class-ph], 2010-2011.
%H <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>
%F Equals 1/A002194. - _Michel Marcus_, Oct 12 2014
%F Equals cosine of the magic angle: cos(A195696). - _Stanislav Sykora_, Mar 07 2016
%F Equals square root of A010701. - _Michel Marcus_, Mar 07 2016
%F Equals 1 + Sum_{k>=0} -(4*k+1)^(-1/2) + (4*k+3)^(-1/2) + (4*k+5)^(-1/2) - (4*k+7)^(-1/2). - _Gerry Martens_, Nov 22 2022
%F Equals (1/2)*(2 - zeta(1/2,1/4) + zeta(1/2,3/4) + zeta(1/2,5/4) - zeta(1/2,7/4)). - _Gerry Martens_, Nov 22 2022
%F Has periodic continued fraction expansion [0, 1; 1, 2] (A040001). - _Michal Paulovic_, Mar 21 2023
%e 0.577350269189625764509148780501957455647601751270126876018602326....
%p evalf(1/sqrt(3)); # _Michal Paulovic_, Mar 21 2023
%t RealDigits[N[1/Sqrt[3],200]] (* _Vladimir Joseph Stephan Orlovsky_, May 27 2010 *)
%o (PARI) \\ Works in v2.15.0; n = 100 decimal places
%o my(n=100); digits(floor(10^n/quadgen(12))) \\ _Michal Paulovic_, Mar 21 2023
%Y Cf. A002194 (sqrt(3)), A010701 (1/3).
%Y Cf. A002193, A165952, A195696, A040001.
%K nonn,cons
%O 0,1
%A _N. J. A. Sloane_
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