Displaying 1-10 of 22 results found.
Egyptian fraction representation of sqrt(12) ( A010469) using a greedy function.
+20
0
3, 3, 8, 174, 47270, 3322246062, 13585339584457844199, 266643312158266377656241697792775202384, 221110316712057155914682414678073188192934894445719392090279403577596961625414
MATHEMATICA
Egyptian[nbr_] := Block[{lst = {IntegerPart[nbr]}, cons = N[ FractionalPart[ nbr], 2^20], denom, iter = 8}, While[ iter >
0, denom = Ceiling[ 1/cons]; AppendTo[ lst, denom]; cons -= 1/denom; iter--]; lst]; Egyptian[ Sqrt[ 12]]
Decimal expansion of sqrt(3).
(Formerly M4326 N1812)
+10
167
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
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 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
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
Kiran S. Kedlaya, A < B, (1999) Problem 6.4, p. 6.
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
EXAMPLE
1.73205080756887729352744634150587236694280525381038062805580697945193...
MATHEMATICA
RealDigits[Sqrt[3], 10, 100][[1]]
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
Decimal expansion of sqrt(3)/2.
+10
86
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
COMMENTS
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
REFERENCES
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.
FORMULA
Equals sin(Pi/3) = cos(Pi/6).
Equals Integral_{x=0..Pi/3} cos(x) dx. (End)
Equals x^(x^(x^...)) where x = (3/4)^(1/sqrt(3)) (infinite power tower). - Michal Paulovic, Jun 25 2023
EXAMPLE
0.86602540378443864676372317...
PROG
(PARI) 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 (Magma) SetDefaultRealField(RealField(100)); Sqrt(3)/2; // G. C. Greubel, Nov 02 2018
EXTENSIONS
Last term corrected and more terms added by Harry J. Smith, Jun 02 2009
Decimal expansion of Pi/(2*sqrt(3)).
+10
41
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
COMMENTS
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
REFERENCES
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.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 30.
FORMULA
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] 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 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 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
EXAMPLE
0.906899682117108925297039128821077866142033124046370287784942...
CROSSREFS
Related constants: A020769, A020789, A093825, A222066, A222067, A222068, A222069, A222070, A222071, A222072, A260646, (1/2)* A093602, A346585, A346584, A346583.
Decimal expansion of 3*(sqrt(25 + 10*sqrt(5))), the surface area of a regular dodecahedron with edges of unit length.
+10
14
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
COMMENTS
Surface area of a regular dodecahedron: A = 3*(sqrt(25 + 10*sqrt(5)))* a^2, where 'a' is the edge.
REFERENCES
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.
FORMULA
Equals 15/tan(Pi/5).
Equals 15*phi/xi, where phi is the golden ratio ( A001622) and xi its associate ( A182007). (End)
EXAMPLE
20.64572880706760307310814372866331519288849004012237995...
MATHEMATICA
RealDigits[3*Sqrt[25+10*Sqrt[5]], 10, 120][[1]] (* Harvey P. Dale, Jun 21 2011 *)
PROG
(PARI) default(realprecision, 100); 3*(sqrt(25 + 10*sqrt(5))) \\ G. C. Greubel, Nov 02 2018 (Magma) SetDefaultRealField(RealField(100)); 3*(Sqrt(25 + 10*Sqrt(5))); // G. C. Greubel, Nov 02 2018
Denominators of continued fraction convergents to sqrt(12).
+10
9
1, 2, 13, 28, 181, 390, 2521, 5432, 35113, 75658, 489061, 1053780, 6811741, 14677262, 94875313, 204427888, 1321442641, 2847313170, 18405321661, 39657956492, 256353060613, 552364077718, 3570537526921
COMMENTS
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
FORMULA
G.f.: (1+2*x-x^2)/(1-14*x^2+x^4). - Colin Barker, Jan 01 2012 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)
MAPLE
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
MATHEMATICA
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 *)
Decimal expansion of the surface index of a regular octahedron.
+10
9
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
COMMENTS
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).
EXAMPLE
5.7191057579816194425444539723965629466374425679...
MATHEMATICA
RealDigits[Sqrt[3]Surd[36, 3], 10, 120][[1]] (* Harvey P. Dale, Mar 12 2015 *)
Numerators of continued fraction convergents to sqrt(12).
+10
7
3, 7, 45, 97, 627, 1351, 8733, 18817, 121635, 262087, 1694157, 3650401, 23596563, 50843527, 328657725, 708158977, 4577611587, 9863382151, 63757904493, 137379191137, 888033051315, 1913445293767, 12368704813917
FORMULA
G.f.: (3+7*x+3*x^2-x^3)/(1-14*x^2+x^4).
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)
MATHEMATICA
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 *)
Decimal expansion of 2*sqrt(3)/(9*Pi).
+10
5
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
COMMENTS
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.)
EXAMPLE
0.122517532315953788780294777402882098...
MATHEMATICA
RealDigits[(2Sqrt[3])/(9Pi), 10, 120][[1]] (* Harvey P. Dale, Nov 17 2013 *)
Decimal expansion of 3+2*sqrt(3).
+10
5
6, 4, 6, 4, 1, 0, 1, 6, 1, 5, 1, 3, 7, 7, 5, 4, 5, 8, 7, 0, 5, 4, 8, 9, 2, 6, 8, 3, 0, 1, 1, 7, 4, 4, 7, 3, 3, 8, 8, 5, 6, 1, 0, 5, 0, 7, 6, 2, 0, 7, 6, 1, 2, 5, 6, 1, 1, 1, 6, 1, 3, 9, 5, 8, 9, 0, 3, 8, 6, 6, 0, 3, 3, 8, 1, 7, 6, 0, 0, 0, 7, 4, 1, 6, 2, 2, 9, 2, 3, 7, 3, 5, 1, 4, 4, 9, 7, 1, 5, 1, 3, 5, 1, 2, 5
COMMENTS
Continued fraction expansion of 3+2*sqrt(3) is A010696 preceded by 6. For a spinning black hole the phase transition to positive specific heat happens at a point governed by 2*sqrt(3)-3 (according to a discussion on John Baez's blog), and not at the golden ratio as claimed by Paul Davis. - Peter Luschny, Mar 02 2013 In particular: a black hole with J > (2*sqrt(3)-3) Gm^2/c has positive specific heat, and negative specific heat if J is less, where J is its angular momentum, m is its mass, G is the gravitational constant, and c is the speed of light. For a solar mass black hole, this is 4.08 * 10^41 joule-seconds or a rotation every 1.61 days with the sun's inertia. - Charles R Greathouse IV, Sep 20 2013
FORMULA
Equals Sum_{n>=1} (sqrt(3)/2)^n = (sqrt(3)/2)/(1 - (sqrt(3)/2)). - Fred Daniel Kline, Mar 03 2014
EXAMPLE
3+2*sqrt(3) = 6.46410161513775458705...
MATHEMATICA
Circs[n_] := With[{r = Sin[Pi/n]/(1 - Sin[Pi/n])}, Graphics[Append[Table[Circle[(r + 1) {Sin[2 Pi k/n], Cos[2 Pi k/n]}, r], {k, n}], {Blue, Circle[{0, 0}, 1]}]]]; Circs[3] (* Charles R Greathouse IV, Jan 14 2013 *)
CROSSREFS
Cf. A002194 (decimal expansion of sqrt(3)), A010469 (decimal expansion of sqrt(12)), A010696 (repeat 2, 6).
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