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 A002194 Decimal expansion of sqrt(3). (Formerly M4326 N1812) 108

%I M4326 N1812

%S 1,7,3,2,0,5,0,8,0,7,5,6,8,8,7,7,2,9,3,5,2,7,4,4,6,3,4,1,5,0,5,8,7,2,

%T 3,6,6,9,4,2,8,0,5,2,5,3,8,1,0,3,8,0,6,2,8,0,5,5,8,0,6,9,7,9,4,5,1,9,

%U 3,3,0,1,6,9,0,8,8,0,0,0,3,7,0,8,1,1,4,6,1,8,6,7,5,7,2,4,8,5,7,5,6,7,5,6,2,6,1,4,1,4,1,5,4

%N Decimal expansion of sqrt(3).

%C "The square root of 3, the 2nd number, after root 2, to be proved irrational, by Theodorus."

%C Length of a diagonal between any vertex of the unit cube and the one corresponding (opposite) vertex not part of the three faces meeting at the original vertex. (Diagonal is hypotenuse of a triangle with sides 1 and sqrt(2)). Hence the diameter of the sphere circumscribed around the unit cube; the ratio of the diameter of any sphere to the edge length of its inscribed cube. - _Rick L. Shepherd_, Jun 09 2005

%C The square root of 3 is the length of the minimal Y-shaped (symmetrical) network linking three points unit distance apart. - _Lekraj Beedassy_, Apr 12 2006

%C Continued fraction expansion is 1 followed by {1, 2} repeated. - _Harry J. Smith_, Jun 01 2009

%C Also, tan(Pi/3) = 2 sin(Pi/3). - _M. F. Hasler_, Oct 27 2011

%C Surface of regular tetrahedron with unit edge. - _Stanislav Sykora_, May 31 2012

%C This is the case n=6 of Gamma(1/n)*Gamma((n-1)/n)/(Gamma(2/n)*Gamma((n-2)/n)) = 2*cos(Pi/n), therefore sqrt(3) = A175379*A203145/(A073005*A073006). - _Bruno Berselli_, Dec 13 2012

%C Ratio of base length to leg length in the isosceles "vampire" triangle, that is, the only isosceles triangle without reflection triangle. The product of cosines of the internal angles of a triangle with sides 1, 1 and sqrt(3) and all similar triangles is -3/8. Hence its reflection triangle is degenerate. See the link below. - _Martin Janecke_, May 09 2013

%C Half of the surface of regular octahedron with unit edge (A010469), and one fifth that of a regular icosahedron with unit edge (i.e., 2*A010527). - _Stanislav Sykora_, Nov 30 2013

%C Diameter of a sphere whose surface area equals 3*Pi. More generally, the square root of x is also the diameter of a sphere whose surface area equals x*Pi. - _Omar E. Pol_, Nov 11 2018

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, England, 1997, page 23.

%H Harry J. Smith, <a href="/A002194/b002194.txt">Table of n, a(n) for n = 1..20000</a>

%H M. F. Jones, <a href="http://www.jstor.org/stable/2004806">22900D approximations to the square roots of the primes less than 100</a>, Math. Comp., 22 (1968), 234-235.

%H Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/sqrt_base">Index of expansions of sqrt(d) in base b</a>

%H R. J. Nemiroff and J. Bonnell, <a href="http://antwrp.gsfc.nasa.gov/htmltest/gifcity/sqrt3.1mil">The first 1 million digits of the square root of 3</a>

%H Simon Plouffe, Plouffe's Inverter, <a href="http://www.plouffe.fr/simon/constants/sqrt3.txt">The square root of 3 to 10 million digits</a>

%H Horace S. Uhler, <a href="http://www.pnas.org/content/37/7/443.full.pdf">Approximations exceeding 1300 decimals for sqrt 3, 1/sqrt 3, sin(pi/3) and distribution of digits in them/a>, Proc. Nat. Acad. Sci. U. S. A. 37, (1951). 443-447.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ReflectionTriangle.html">Reflection Triangle</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareRoot.html">Square Root</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TheodorussConstant.html">Theodorus's Constant</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Platonic solid">Platonic solid</a>

%e 1.73205080756887729352744634150587236694280525381038062805580697945193...

%t RealDigits[Sqrt, 10, 100][]

%o (PARI) { default(realprecision, 20080); x=(sqrt(3)); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002194.txt", n, " ", d)); } \\ _Harry J. Smith_, Jun 01 2009

%o (MAGMA) SetDefaultRealField(RealField(100)); Sqrt(3); // _G. C. Greubel_, Aug 21 2018

%Y Cf. A040001 (continued fraction).

%Y Cf. A010469 (double), A010527 (half), A131595 (surface of regular dodecahedron). - _Stanislav Sykora_, Nov 30 2013

%K cons,nonn,easy

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Robert G. Wilson v_, Dec 07 2000

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Last modified November 13 23:48 EST 2019. Contains 329106 sequences. (Running on oeis4.)