A002194 Decimal expansion of sqrt(3).

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• cons
• nonn
• easy
• changed

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

Examples

			1.73205080756887729352744634150587236694280525381038062805580697945193...
		

References

• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 24, 184.
• Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.4 Irrational Numbers and §12.4 Theorems and Formulas (Solid Geometry), pp. 84, 450.
• 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
• Madeleine Bonsma-Fisher and Kent Bonsma-Fisher, How big a table do you need for your jigsaw puzzle?, arXiv:2312.04588 [math.HO], 2023.
• 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.
• Horace S. Uhler, Approximations exceeding 1300 decimals for sqrt 3, 1/sqrt 3, sin(pi/3) and distribution of digits in them, Proc. Nat. Acad. Sci. U. S. A., Vol. 37, No. 7 (1951), pp. 443-447.
• 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.
• Index entries for algebraic numbers, degree 2

Crossrefs

Cf. A040001 (continued fraction), A220335.

Cf. A010469 (double), A010527 (half), A131595 (surface of regular dodecahedron).

Programs

Magma

SetDefaultRealField(RealField(100)); Sqrt(3); // G. C. Greubel, Aug 21 2018

Maple

evalf(sqrt(3), 100); # Michal Paulovic, Feb 24 2023

Mathematica

RealDigits[Sqrt[3], 10, 100][[1]]

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

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

Equals Product_{n>=1} 3^(1/3^n). - Michal Paulovic, Feb 24 2023

Equals Product_{n>=0} ((6*n + 2)*(6*n + 4))/((6*n + 1)*(6*n + 5)). - Antonio Graciá Llorente, Feb 22 2024

Equals tan(Pi/3) = A010527/(1/2). - R. J. Mathar, Aug 31 2025

Extensions

More terms from Robert G. Wilson v, Dec 07 2000