|
%I M4326 N1812 #139 Feb 24 2024 11:03:30
%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
%C Sometimes called Theodorus's constant, after the ancient Greek mathematician Theodorus of Cyrene (5th century BC). - _Amiram Eldar_, Apr 02 2022
%C For any triangle ABC, cotan(A) + cotan(B) + cotan(C) >= sqrt(3); equality is obtained only when the triangle is equilateral (see the Kiran S. Kedlaya link). - _Bernard Schott_, Sep 13 2022
%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 Madeleine Bonsma-Fisher and Kent Bonsma-Fisher, <a href="https://arxiv.org/abs/2312.04588">How big a table do you need for your jigsaw puzzle?</a>, arXiv:2312.04588 [math.HO], 2023.
%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., Vol. 22, No. 101 (1968), pp. 234-235.
%H Kiran S. Kedlaya, <a href="https://igor-kortchemski.perso.math.cnrs.fr/olympiades/Cours/ineqs-080299.pdf">A < B</a>, (1999) Problem 6.4, p. 6.
%H Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/sqrt_base">Index of expansions of sqrt(d) in base b</a>.
%H Robert J. Nemiroff and Jerry 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="https://doi.org/10.1073/pnas.37.7.443">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., Vol. 37, No. 7 (1951), pp. 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>.
%H <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>
%F Equals Sum_{k>=0} binomial(2*k,k)/6^k = Sum_{k>=0} binomial(2*k,k) * k/6^k. - _Amiram Eldar_, Aug 03 2020
%F sqrt(3) = 1 + 1/2 + 1/(2*3) + 1/(2*3*4) + 1/(2*3*4*2) + 1/(2*3*4*2*8) + 1/(2*3*4*2*8*14) + 1/(2*3*4*2*8*14*2) + 1/(2*3*4*2*8*14*2*98) + 1/(2*3*4*2*8*14*2*98*194) + .... (Define F(n) = (n-1)*sqrt(n^2 - 1) - (n^2 - n - 1). Show F(n) = 1/2 + 1/(2*(n+1)) + 1/(2*(n+1)*(2*n)) + 1/(2*(n+1)*(2*n))*F(2*n^2 - 1) for n >= 0; then iterate this identity at n = 2. See A220335.) - _Peter Bala_, Mar 18 2022
%F Equals i^(1/3) + i^(-1/3). - _Gary W. Adamson_, Jul 06 2022
%F Equals Product_{n>=1} 3^(1/3^n). - _Michal Paulovic_, Feb 24 2023
%F Equals Product_{n>=0} ((6*n + 2)*(6*n + 4))/((6*n + 1)*(6*n + 5)). - _Antonio Graciá Llorente_, Feb 22 2024
%e 1.73205080756887729352744634150587236694280525381038062805580697945193...
%p evalf(sqrt(3), 100); # _Michal Paulovic_, Feb 24 2023
%t RealDigits[Sqrt[3], 10, 100][[1]]
%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), A220335.
%Y Cf. A010469 (double), A010527 (half), A131595 (surface of regular dodecahedron).
%K cons,nonn,easy
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Robert G. Wilson v_, Dec 07 2000
|
