A010469 -id:A010469 - cp's OEIS frontend

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

N. J. A. Sloane

"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

			1.73205080756887729352744634150587236694280525381038062805580697945193...

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.

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

Cf. A040001 (continued fraction), A220335.

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

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

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

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

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

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

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

A010527 Decimal expansion of sqrt(3)/2.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

This is the ratio of the height of an equilateral triangle to its base.
Essentially the same sequence arises from decimal expansion of square root of 75, which is 8.6602540378443864676372317...
Also the real part of i^(1/3), the cubic root of i. - Stanislav Sykora, Apr 25 2012
Gilbert & Pollak conjectured that this is the Steiner ratio rho_2, the least upper bound of the ratio of the length of the Steiner minimal tree to the length of the minimal tree in dimension 2. (See Ivanov & Tuzhilin for the status of this conjecture as of 2012.) - Charles R Greathouse IV, Dec 11 2012
Surface area of a regular icosahedron with unit edge is 5*sqrt(3), i.e., 10 times this constant. - Stanislav Sykora, Nov 29 2013
Circumscribed sphere radius for a cube with unit edges. - Stanislav Sykora, Feb 10 2014
Also the ratio between the height and the pitch, used in the Unified Thread Standard (UTS). - Enrique Pérez Herrero, Nov 13 2014
Area of a 30-60-90 triangle with shortest side equal to 1. - Wesley Ivan Hurt, Apr 09 2016
If a, b, c are the sides of a triangle ABC and h_a, h_b, h_c the corresponding altitudes, then (h_a+h_b+h_c) / (a+b+c) <= sqrt(3)/2; equality is obtained only when the triangle is equilateral (see Mitrinovic reference). - Bernard Schott, Sep 26 2022

			0.86602540378443864676372317...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 8.2, 8.3 and 8.6, pp. 484, 489, and 504.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), pp. 450-451.
D. S. Mitrinovic, E. S. Barnes, D. C. B. Marsh, and J. R. M. Radok, Elementary Inequalities, Tutorial Text 1 (1964), P. Noordhoff LTD, Groningen, problem 6.8, page 114.

Harry J. Smith, Table of n, a(n) for n = 0..20000
E. N. Gilbert and H. O. Pollak, Steiner minimal trees, SIAM J. Appl. Math. 16, (1968), pp. 1-29.
A. O. Ivanov and A. A. Tuzhilin, The Steiner ratio Gilbert-Pollak conjecture is still open, Algorithmica 62:1-2 (2012), pp. 630-632.
Matt Parker, The mystery of 0.866025403784438646763723170752936183471402626905190314027903489, Stand-up Maths, YouTube video, Feb 14 2024.
Simon Plouffe, Plouffe's Inverter, sqrt(3)/2 to 10000 digits.
Simon Plouffe, Sqrt(3)/2 to 5000 digits.
Eric Weisstein's World of Mathematics, Lebesgue Minimal Problem.
Wikipedia, Icosahedron.
Wikipedia, Platonic solid.
Wikipedia, Unified Thread Standard.
Index entries for algebraic numbers, degree 2.

Cf. A010153.
Cf. Platonic solids surfaces: A002194 (tetrahedron), A010469 (octahedron), A131595 (dodecahedron).
Cf. Platonic solids circumradii: A010503 (octahedron), A019881 (icosahedron), A179296 (dodecahedron), A187110 (tetrahedron).
Cf. A126664 (continued fraction), A144535/A144536 (convergents).
Cf. A002194, A010502, A020821, A104956, A152623 (other geometric inequalities).

SetDefaultRealField(RealField(100)); Sqrt(3)/2; // G. C. Greubel, Nov 02 2018

Digits:=100: evalf(sqrt(3)/2); # Wesley Ivan Hurt, Apr 09 2016

RealDigits[Sqrt[3]/2, 10, 200][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

default(realprecision, 20080); x=10*(sqrt(3)/2); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b010527.txt", n, " ", d));  \\ Harry J. Smith, Jun 02 2009

sqrt(3)/2 \\ Michel Marcus, Apr 10 2016

Equals cos(30 degrees). - Kausthub Gudipati, Aug 15 2011
Equals A002194/2. - Stanislav Sykora, Nov 30 2013
From Amiram Eldar, Jun 29 2020: (Start)
Equals sin(Pi/3) = cos(Pi/6).
Equals Integral_{x=0..Pi/3} cos(x) dx. (End)
Equals 1/(10*A020832). - Bernard Schott, Sep 29 2022
Equals x^(x^(x^...)) where x = (3/4)^(1/sqrt(3)) (infinite power tower). - Michal Paulovic, Jun 25 2023
Equals 2F1(-1/4,1/4 ; 1/2 ; 3/4) . - R. J. Mathar, Aug 31 2025

Last term corrected and more terms added by Harry J. Smith, Jun 02 2009

A093766 Decimal expansion of Pi/(2*sqrt(3)).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Apr 15 2004

Density of densest packing of equal circles in two dimensions (achieved for example by the A2 lattice).
The number gives the areal coverage (90.68... percent) of the close hexagonal (densest) packing of circles in the plane. The hexagonal unit cell is a rhombus of side length 1 and height sqrt(3)/2; the area of the unit cell is sqrt(3)/2 and the four parts of circles add to an area of one circle of radius 1/2, which is Pi/4. - R. J. Mathar, Nov 22 2011
Ratio of surface area of a sphere to the regular octahedron whose edge equals the diameter of the sphere. - Omar E. Pol, Dec 09 2013

			0.906899682117108925297039128821077866142033124046370287784942...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer, 3rd. ed., 1998. See p. xix.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.7, p. 506.
L. B. W. Jolley, Summation of Series, Dover (1961), Eq. (84) on page 16.
Joel L. Schiff, The Laplace Transform: Theory and Applications, Springer-Verlag New York, Inc. (1999). See p. 149.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 30.

J. H. Conway and N. J. A. Sloane, What are all the best sphere packings in low dimensions?, Discr. Comp. Geom., 13 (1995), 383-403.
Xi Lin, Dirk Schmelter, Sadaf Imanian, and Horst Hintze-Bruening, Hierarchically Ordered alpha-Zirconium Phosphate Platelets in Aqueous Phase with Empty Liquid, Scientific Reports (2019) Vol. 9, Article No. 16389.
R. J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015. See Table 22 for L(m=6,r=2,s=1).
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
Michael I. Shamos, A catalog of the real numbers, (2007). See pp. 625-626.
N. J. A. Sloane and Andrey Zabolotskiy, Table of maximal density of a packing of equal spheres in n-dimensional Euclidean space (some values are only conjectural).
Eckard Specht, May 21 2012, The best known packings of equal circles in a circle (complete up to N=1500).
László Fejes Tóth, An Inequality concerning polyhedra, Bull. Amer. Math. Soc. 54 (1948), 139-146. See p. 146.
Eric Weisstein's World of Mathematics, Smoothed Octagon.
Eric Weisstein's World of Mathematics, Circle Packing.
Index entries for transcendental numbers.

Related constants: A020769, A020789, A093825, A222066, A222067, A222068, A222069, A222070, A222071, A222072, A260646, (1/2)*A093602, A346585, A346584, A346583.

Cf. A072691, A381671.

RealDigits[Pi/(2 Sqrt[3]), 10, 111][[1]] (* Robert G. Wilson v, Nov 07 2012 *)

Pi/sqrt(12) \\ Charles R Greathouse IV, Oct 31 2014

Equals (5/6)*(7/6)*(11/12)*(13/12)*(17/18)*(19/18)*(23/24)*(29/30)*(31/30)*..., where the numerators are primes > 3 and the denominators are the nearest multiples of 6.
Equals Sum_{n>=1} 1/A134667(n). [Jolley]
Equals Sum_{n>=0} (-1)^n/A124647(n). [Jolley eq. 273]
Equals A000796 / A010469. - Omar E. Pol, Dec 09 2013
Continued fraction expansion: 1 - 2/(18 + 12*3^2/(24 + 12*5^2/(32 + ... + 12*(2*n - 1)^2/((8*n + 8) + ... )))). See A254381 for a sketch proof. - Peter Bala, Feb 04 2015
From Peter Bala, Feb 16 2015: (Start)
Equals 4*Sum_{n >= 0} 1/((6*n + 1)*(6*n + 5)).
Continued fraction: 1/(1 + 1^2/(4 + 5^2/(2 + 7^2/(4 + 11^2/(2 + ... + (6*n + 1)^2/(4 + (6*n + 5)^2/(2 + ... ))))))). (End)
The inverse is (2*sqrt(3))/Pi = Product_{n >= 1} 1 + (1 - 1/(4*n))/(4*n*(9*n^2 - 9*n + 2)) = (35/32) * (1287/1280) * (8075/8064) * (5635/5632) * (72819/72800) * ... =  1.102657790843585... - Dimitris Valianatos, Aug 31 2019
From Amiram Eldar, Aug 15 2020: (Start)
Equals Integral_{x=0..oo} 1/(x^2 + 3) dx.
Equals Integral_{x=0..oo} 1/(3*x^2 + 1) dx. (End)
Equals 1 + Sum_{k>=1} ( 1/(6*k+1) - 1/(6*k-1) ). - Sean A. Irvine, Jul 24 2021
For positive integer k, Pi/(2*sqrt(3)) = Sum_{n >= 0} (6*k + 4)/((6*n + 1)*(6*n + 6*k + 5)) - Sum_{n = 0..k-1} 1/(6*n + 5). - Peter Bala, Jul 10 2024
From Stefano Spezia, Jun 05 2025: (Start)
Equals Sum_{k>=0} (-1)^k/((k + 1)*(3*k + 1)).
Equals Integral_{x=0..oo} 1/(x^4 + x^2 + 1) dx.
Equals Integral_{x=0..oo} x^2/(x^4 + x^2 + 1) dx. (End)
Equals sqrt(A072691) = 3*A381671. - Hugo Pfoertner, Jun 05 2025

Entry revised by N. J. A. Sloane, Feb 10 2013

A131595 Decimal expansion of 3(sqrt(25 + 10sqrt(5))), the surface area of a regular dodecahedron with edges of unit length.

Original entry on oeis.org

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

Offset: 2

Omar E. Pol, Aug 30 2007

Surface area of a regular dodecahedron: A = 3*(sqrt(25 + 10*sqrt(5)))* a^2, where 'a' is the edge.

			20.64572880706760307310814372866331519288849004012237995...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.

G. C. Greubel, Table of n, a(n) for n = 2..10001
Eric Weisstein's World of Mathematics, Dodecahedron.
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 4.

Cf. A102769, A001622 (phi), A182007 (associate of phi), A010527 (icosahedron/10), A010469 (octahedron), A002194 (tetrahedron). - Stanislav Sykora, Nov 30 2013

SetDefaultRealField(RealField(100)); 3*(Sqrt(25 + 10*Sqrt(5))); // G. C. Greubel, Nov 02 2018

evalf(3*(sqrt(25+10*sqrt(5))),130); # Muniru A Asiru, Nov 02 2018

RealDigits[3*Sqrt[25+10*Sqrt[5]],10,120][[1]] (* Harvey P. Dale, Jun 21 2011 *)

default(realprecision, 100); 3*(sqrt(25 + 10*sqrt(5))) \\ G. C. Greubel, Nov 02 2018

From Stanislav Sykora, Nov 30 2013: (Start)
Equals 15/tan(Pi/5).
Equals 15*phi/xi, where phi is the golden ratio (A001622) and xi its associate (A182007). (End)

More terms from Harvey P. Dale, Jun 21 2011

A041017 Denominators of continued fraction convergents to sqrt(12).

Original entry on oeis.org

1, 2, 13, 28, 181, 390, 2521, 5432, 35113, 75658, 489061, 1053780, 6811741, 14677262, 94875313, 204427888, 1321442641, 2847313170, 18405321661, 39657956492, 256353060613, 552364077718, 3570537526921

Offset: 0

N. J. A. Sloane

a(2n+1)/a(2n) tends to 1/(sqrt(12) - 3) = 2.154700538...; e.g., a(7)/a(6) = 5432/2521 = 2.1547005...; but a(2n)/a(2n - 1) tends to 6.464101615... = sqrt(12) + 3; e.g., a(8)/a(7) = 35113/5432 = 6.46101620... - Gary W. Adamson, Mar 28 2004
The constant sqrt(12) + 3 = 6.464101615... is the "curvature" (reciprocal of the radius) of the inner or 4th circle in the Descartes circle equation; given 3 mutually tangent circles of radius 1, the radius of the innermost tangential circle = 0.1547005383... = 1/(sqrt(12) + 3). The Descartes circle equation states that given 4 mutually tangent circles (i.e., 3 tangential plus the innermost circle) with curvatures a,b,c,d (curvature = 1/r), then (a^2 + b^2 + c^2 + d^2) = 1/2(a + b + c + d)^2. - Gary W. Adamson, Mar 28 2004
Sequence also gives numerators in convergents to barover[6,2] = CF: [6,2,6,2,6,2,...] = 0.1547005... = 1/(sqrt(12) + 3), the first few convergents being 1/6, 2/13, 13/84, 28/181, 181/1170, 390/2521... with 390/2521 = 0.154700515... - Gary W. Adamson, Mar 28 2004
Sqrt(12) = 3 + continued fraction [2, 6, 2, 6, 2, 6, ...] = 6/2 + 6/13 + 6/(13*181) + 6/(181*2521) + ... - Gary W. Adamson, Dec 21 2007
Also, values i where A227790(i)/i reaches a new maximum (conjectured). - Ralf Stephan, Sep 23 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,14,0,-1).

Cf. A010469, A040008, A041016 (numerators).

with (numtheory): seq( nthdenom(cfrac(sin(Pi/6)*tan(Pi/3),25),i)-nthnumer(cfrac(sin(Pi/6)*tan(Pi/3),25),i), i=2..24 ); # Zerinvary Lajos, Feb 10 2007

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[12],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011 *)
Denominator[Convergents[Sqrt[12],50]] (* Harvey P. Dale, Feb 18 2012 *)
a0[n_] := ((7-4*Sqrt[3])^n*(2+Sqrt[3])-(-2+Sqrt[3])*(7+4*Sqrt[3])^n)/4 // Simplify
a1[n_] := 2*Sum[a0[i], {i, 1, n}]
Flatten[MapIndexed[{a0[#],a1[#]}&,Range[11]]] (* Gerry Martens, Jul 10 2015 *)

G.f.: (1+2*x-x^2)/(1-14*x^2+x^4). - Colin Barker, Jan 01 2012
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)]:
a0(n) = ((7-4*sqrt(3))^n*(2+sqrt(3)) - (-2+sqrt(3))*(7+4*sqrt(3))^n)/4.
a1(n) = 2*Sum_{i=1..n} a0(i). (End)

A232811 Decimal expansion of the surface index of a regular octahedron.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Dec 01 2013

Equivalently, the surface area of a regular octahedron with unit volume. Among Platonic solids, surface indices decrease with increasing number of faces: A232812 (tetrahedron), 6.0 (cube = hexahedron), this one, A232810 (dodecahedron), and A232809 (icosahedron).

An algebraic integer of degree 6 with minimal polynomial x^6 - 34992. - Charles R Greathouse IV, Apr 25 2016

			5.7191057579816194425444539723965629466374425679...

Stanislav Sykora, Table of n, a(n) for n = 1..1000
Wikipedia, Platonic solid
Index entries for algebraic numbers, degree 6

Cf. A010469, A131594, A232808 (surface index for a sphere), A232809, A232810, A232812.

RealDigits[Sqrt[3]Surd[36,3],10,120][[1]] (* Harvey P. Dale, Mar 12 2015 *)

sqrtn(34992,6) \\ Charles R Greathouse IV, Apr 25 2016

sqrt(3)*6^(2/3).

A010469/A131594^(2/3).

A041016 Numerators of continued fraction convergents to sqrt(12).

Original entry on oeis.org

3, 7, 45, 97, 627, 1351, 8733, 18817, 121635, 262087, 1694157, 3650401, 23596563, 50843527, 328657725, 708158977, 4577611587, 9863382151, 63757904493, 137379191137, 888033051315, 1913445293767, 12368704813917

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,14,0,-1).

Cf. A010469, A041017.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[12],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011*)
Numerator[Convergents[Sqrt[12], 30]] (* Vincenzo Librandi, Oct 28 2013 *)
a0[n_] := (-((7-4*Sqrt[3])^n*(3+2*Sqrt[3]))+(-3+2*Sqrt[3])*(7+4*Sqrt[3])^n)/2 //Simplify
a1[n_] := ((7-4*Sqrt[3])^n+(7+4*Sqrt[3])^n)/2 // Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &,Range[20]]] (* Gerry Martens, Jul 11 2015 *)
LinearRecurrence[{0,14,0,-1},{3,7,45,97},30] (* Harvey P. Dale, Jun 02 2016 *)

G.f.: (3+7*x+3*x^2-x^3)/(1-14*x^2+x^4).
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
a0(n) = (-((7-4*sqrt(3))^n*(3+2*sqrt(3)))+(-3+2*sqrt(3))*(7+4*sqrt(3))^n)/2.
a1(n) = ((7-4*sqrt(3))^n+(7+4*sqrt(3))^n)/2. (End)

A040008 Continued fraction for sqrt(12).

Original entry on oeis.org

3, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6

Offset: 0

N. J. A. Sloane

Eventual period is (2,6). - Zak Seidov, Mar 05 2011

Decimal expansion of 323/990. - R. J. Mathar, Aug 22 2025

			3.464101615137754587054892683... = 3 + 1/(2 + 1/(6 + 1/(2 + 1/(6 + ...)))). - _Harry J. Smith_, Jun 02 2009

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010469 Decimal expansion, A010696.

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[12],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)

{ allocatemem(932245000); default(realprecision, 24000); x=contfrac(sqrt(12)); for (n=0, 20000, write("b040008.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 02 2009

G.f.: (3 + 2*x + 3*x^2)/(1 - x^2). - Stefano Spezia, Jul 26 2025

A165922 Decimal expansion of 2sqrt(3)/(9Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Sep 30 2009

The ratio of the volume of a regular tetrahedron to the volume of the circumscribed sphere. (The MathWorld link shows that the circumradius for a tetrahedron with side length a is a*sqrt(6)/4.)

			0.122517532315953788780294777402882098...

Frank Jackson and Eric W. Weisstein, Tetrahedron
Index entries for transcendental numbers

Cf. A000796, A002194, A010469, A049541, A020784.

RealDigits[(2Sqrt[3])/(9Pi),10,120][[1]] (* Harvey P. Dale, Nov 17 2013 *)

PARI
```
2*3^(-3/2)/Pi
```

2*sqrt(3)/(9*Pi) = A010469/(9*A000796) = (2/9)*A002194/A000796 = (2/9)*A002194*A049541 = 2*A020784/A000796 = 2*3^(-3/2)/Pi.

cp's OEIS Frontend

A248241 Egyptian fraction representation of sqrt(12) (A010469) using a greedy function.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A002194 Decimal expansion of sqrt(3).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A010527 Decimal expansion of sqrt(3)/2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

Extensions

A093766 Decimal expansion of Pi/(2*sqrt(3)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A131595 Decimal expansion of 3*(sqrt(25 + 10*sqrt(5))), the surface area of a regular dodecahedron with edges of unit length.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A041017 Denominators of continued fraction convergents to sqrt(12).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

A131595 Decimal expansion of 3(sqrt(25 + 10sqrt(5))), the surface area of a regular dodecahedron with edges of unit length.

A165922 Decimal expansion of 2sqrt(3)/(9Pi).