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 A002194 Decimal expansion of sqrt(3). (Formerly M4326 N1812) 133
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS "The square root of 3, the 2nd number, after root 2, to be proved irrational, by Theodorus." 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 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 Continued fraction expansion is 1 followed by {1, 2} repeated. - Harry J. Smith, Jun 01 2009 Also, tan(Pi/3) = 2 sin(Pi/3). - M. F. Hasler, Oct 27 2011 Surface of regular tetrahedron with unit edge. - Stanislav Sykora, May 31 2012 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 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 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 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 Sometimes called Theodorus's constant, after the ancient Greek mathematician Theodorus of Cyrene (5th century BC). - Amiram Eldar, Apr 02 2022 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 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, page 23. LINKS Harry J. Smith, Table of n, a(n) for n = 1..20000 M. F. Jones, 22900D approximations to the square roots of the primes less than 100, Math. Comp., Vol. 22, No. 101 (1968), pp. 234-235. Kiran S. Kedlaya, A < B, (1999) Problem 6.4, p. 6. Jason Kimberley, Index of expansions of sqrt(d) in base b. Robert J. Nemiroff and Jerry Bonnell, The first 1 million digits of the square root of 3. Simon Plouffe, Plouffe's Inverter, The square root of 3 to 10 million digits. Eric Weisstein's World of Mathematics, Reflection Triangle. Eric Weisstein's World of Mathematics, Square Root. Eric Weisstein's World of Mathematics, Theodorus's Constant. Wikipedia, Platonic solid. FORMULA 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 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 Equals i^(1/3) + i^(-1/3). - Gary W. Adamson, Jul 06 2022 EXAMPLE 1.73205080756887729352744634150587236694280525381038062805580697945193... MATHEMATICA RealDigits[Sqrt, 10, 100][] PROG (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 (Magma) SetDefaultRealField(RealField(100)); Sqrt(3); // G. C. Greubel, Aug 21 2018 CROSSREFS Cf. A040001 (continued fraction), A220335. Cf. A010469 (double), A010527 (half), A131595 (surface of regular dodecahedron). Sequence in context: A204155 A355673 A160390 * A355674 A256843 A033327 Adjacent sequences:  A002191 A002192 A002193 * A002195 A002196 A002197 KEYWORD cons,nonn,easy,changed AUTHOR EXTENSIONS More terms from Robert G. Wilson v, Dec 07 2000 STATUS approved

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Last modified October 1 16:19 EDT 2022. Contains 357149 sequences. (Running on oeis4.)