A020832 -id:A020832 - cp's OEIS frontend

A010527 Decimal expansion of sqrt(3)/2.

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

A047235 Numbers that are congruent to {2, 4} mod 6.

Original entry on oeis.org

2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50, 52, 56, 58, 62, 64, 68, 70, 74, 76, 80, 82, 86, 88, 92, 94, 98, 100, 104, 106, 110, 112, 116, 118, 122, 124, 128, 130, 134, 136, 140, 142, 146, 148, 152, 154, 158, 160, 164, 166, 170, 172, 176, 178, 182, 184, 188, 190, 194, 196, 200, 202, 206

Offset: 1

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 19 ).
Complement of A047273; A093719(a(n)) = 0. - Reinhard Zumkeller, Oct 01 2008
One could prefix an initial term "1" (or not) and define this sequence through a(n+1) = a(n) + (a(n) mod 6). See A001651 for the analog with 3, A235700 (with 5), A047350 (with 7), A007612 (with 9) and A102039 (with 10). Using 4 or 8 yields a constant sequence from that term on. - M. F. Hasler, Jan 14 2014
Nonnegative m such that m^2/6 + 1/3 is an integer. - Bruno Berselli, Apr 13 2017
Numbers divisible by 2 but not by 3. - David James Sycamore, Apr 04 2018
Numbers k for which A276086(k) is of the form 6m+3. - Antti Karttunen, Dec 03 2022

David A. Corneth, Table of n, a(n) for n = 1..10000
Chunhui Lai, A note on potentially K_4-e graphical sequences, arXiv:math/0308105 [math.CO], 2003.
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A020760, A020832, A093719, A047273 (complement), A120325 (characteristic function).
Equals 2*A001651.
Cf. A007310 ((6*n+(-1)^n-3)/2). - Bruno Berselli, Jun 24 2010
Positions of 3's in A053669 and in A358840.

[ n eq 1 select 2 else Self(n-1)+2*(1+n mod 2): n in [1..70] ]; // Klaus Brockhaus, Dec 13 2008

seq(6*floor((n+1)/2) + 3 + (-1)^n, n=1..67); # Gary Detlefs, Mar 02 2010

Flatten[Table[{6n - 4, 6n - 2}, {n, 40}]] (* Alonso del Arte, Oct 27 2014 *)

a(n)=(n-1)\2*6+3+(-1)^n \\ Charles R Greathouse IV, Jul 01 2013

first(n) = my(v = vector(n, i, 3*i - 1)); forstep(i = 2, n, 2, v[i]--); v \\ David A. Corneth, Oct 20 2017

a(n) = 2*A001651(n).
n such that phi(3*n) = phi(2*n). - Benoit Cloitre, Aug 06 2003
G.f.: 2*x*(1 + x + x^2)/((1 + x)*(1 - x)^2). a(n) = 3*n - 3/2 - (-1)^n/2. - R. J. Mathar, Nov 22 2008
a(n) = 3*n + 5..n odd, 3*n + 4..n even a(n) = 6*floor((n+1)/2) + 3 + (-1)^n. - Gary Detlefs, Mar 02 2010
a(n) = 6*n - a(n-1) - 6 (with a(1) = 2). - Vincenzo Librandi, Aug 05 2010
a(n+1) = a(n) + (a(n) mod 6). - M. F. Hasler, Jan 14 2014
Sum_{n>=1} 1/a(n)^2 = Pi^2/27. - Dimitris Valianatos, Oct 10 2017
a(n) = (6*n - (-1)^n - 3)/2. - Ammar Khatab, Aug 23 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(6*sqrt(3)). - Amiram Eldar, Dec 11 2021
E.g.f.: 2 + ((6*x - 3)*exp(x) - exp(-x))/2. - David Lovler, Aug 25 2022
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2/sqrt(3) (10 * A020832).
Product_{n>=1} (1 + (-1)^n/a(n)) = 1/sqrt(3) (A020760). (End)

A246724 Decimal expansion of r_2, the second smallest radius for which a compact packing of the plane exists, with disks of radius 1 and r_2.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 02 2014

Essentially the same digit sequence as A176053 and A020832. - R. J. Mathar, Sep 06 2014
This equals the ratio of the radius of the inner Soddy circle and the common radius of the three kissing circles. See A343235, also for links. - Wolfdieter Lang, Apr 19 2021
Previous comment is, together with A176053, the answer to the 1st problem proposed during the 4th Canadian Mathematical Olympiad in 1972. - Bernard Schott, Mar 16 2022

			0.154700538379251529018297561003914911295203502540253752...

Michael Doob, The Canadian Mathematical Olympiad & L'Olympiade Mathématique du Canada 1969-1993 - Canadian Mathematical Society & Société Mathématique du Canada, Problem 1, 1972, page 37, 1993.

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2021, p. 62.
The IMO Compendium, Problem 1, 4th Canadian Mathematical Olympiad, 1972.
Samuel G. Moreno and Esther M. García, New infinite products of cosines and Viète-like formulae, Mathematics Magazine, Vol. 86, No. 1 (2013), pp. 15-25.
Bernard Schott, Soddy circles.
Index entries for algebraic numbers, degree 2.
Index to sequences related to Olympiads.

Cf. A246723 (r_1), A246725 (r_3), A246726 (r_4), A246727 (r_5), A002193 (r_6 = sqrt(2)-1), A246728 (r_7), A246729 (r_8), A246730 (r_9).

Cf. A176053, A343235.

RealDigits[(2*Sqrt[3] - 3)/3, 10, 103] // First

2/sqrt(3) - 1 \\ Charles R Greathouse IV, Feb 10 2025

Equals (2*sqrt(3) - 3)/3.
Equals A176053 - 2.
Equals -1 + sqrt(2) * sqrt(2-sqrt(2)) * sqrt(2-sqrt(2-sqrt(2))) * ... (Moreno and García, 2013). - Amiram Eldar, Jun 09 2022

A089806 Expansion of Jacobi theta function (theta_3(q^(1/3))-theta_2(q^3))/2/q^(1/12).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Eric W. Weisstein, Nov 12 2003

			1 + q^2 + q^4 + q^10 + q^14 + q^24 + q^30 + q^44 + q^52 + ...

Antti Karttunen, Table of n, a(n) for n = 0..10034
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
I. J. Zucker, Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums, J. Phys. A: Math. Gen. 23, 117-132, 1990.
Index entries for characteristic functions.

Cf. A080995(n) = a(2n).

Cf. A020832.

nmax = 100; CoefficientList[Series[Product[(1+x^(2*k)) * (1-x^(6*k)) / (1+x^(6*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 05 2016 *)
Table[If[IntegerQ[Sqrt[12*n + 1]], 1, 0], {n, 0, 100}] (* Vaclav Kotesovec, Dec 29 2023 *)

a(n)=issquare(12*n+1) /* Michael Somos, Apr 13 2005 */

lista(nn) = {q='q+O('q^nn); Vec(eta(q^4)*eta(q^6)^2/(eta(q^2)*eta(q^12)))} \\ Altug Alkan, Mar 22 2018

Euler transform of period 12 sequence [0, 1, 0, 0, 0, -1, 0, 0, 0, 1, 0, -1, ...]. - Michael Somos, Apr 13 2005
a(n) = b(12n+1) where b(n) is multiplicative and b(3^e)=0^e, b(p^e)=(1+(-1)^e)/2 if p<>3. - Michael Somos, Jun 06 2005
Expansion of q^(-1/12)(eta(q^4)eta(q^6)^2)/(eta(q^2)eta(q^12)) in powers of q.
Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 2/sqrt(3) = 1.1547005... (10 * A020832). - Amiram Eldar, Dec 29 2023

A165952 Decimal expansion of 2sqrt(3)/(3Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Oct 02 2009

The ratio of the volume of a cube to the volume of the circumscribed sphere (which has circumradius a*sqrt(3)/2 = a*A010527, where a is the cube's edge length; see MathWorld link). For similar ratios for other Platonic solids, see A165922, A049541, A165953, and A165954. A063723 shows the order of these by size.

			0.3675525969478613663408843322086462942649243202444271018662440135273535356...

Eric Weisstein's World of Mathematics, Cube.
Index entries for transcendental numbers

Cf. A000796, A002194, A165922, A049541, A165953, A165954, A063723, A020760, A020832.

RealDigits[(2*Sqrt[3])/(3Pi),10,120][[1]] (* Harvey P. Dale, Oct 08 2012 *)

PARI
```
2*sqrt(3)/(3*Pi)
```

2*sqrt(3)/(3*Pi) = 2*A002194/(3*A000796) = 3*A165922 = (2*sqrt(3)/3)*A049541 = 10*A020832*A049541 = 2*A020760*A049541.

A181152 Decimal expansion of Madelung constant (negated) for the CsCl structure.

Original entry on oeis.org

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

Offset: 1

Leslie Glasser, Jan 24 2011

This is often quoted for a different lattice constant and multiplied by 2/sqrt(3) = 1.1547... = 10*A020832, which gives 1.76267...*1.1547... = 2.03536151... given in Zucker's Table 5 as the alpha for the CsCl structure, and by Sakamoto as the M_d for the B2 lattice. Given Zucker's b(1) = 0.774386141424002815... = A185577, this constant here is sqrt(3)*(3*b(1)+A085469)/4. - R. J. Mathar, Jan 28 2011

The CsCl structure consists of two interpenetrating simple cubic lattices of ions with charges +1 and -1, together occupying all the sites of the body-centered cubic lattice. - Andrey Zabolotskiy, Oct 21 2019

Leslie Glasser, Solid-State Energetics and Electrostatics: Madelung Constants and Madelung Energies, Inorg. Chem., 2012, 51 (4), 2420-2424.
Y. Sakamoto, Madelung Constants of Simple Crystals Expressed in Terms of Born's Basic Potentials of 15 Figures, Journal of Chemical Physics, 28 (1958), 164-165. Errata: J. Chem. Phys, 28 (1958), 733; J. Chem. Phys, 28 (1958), 1253.
Nicolas Tavernier, Gian Luigi Bendazzoli, Véronique Brumas, Stefano Evangelisti, and J. A. Berger, Clifford boundary conditions: a simple direct-sum evaluation of Madelung constants, J. Phys. Chem. Lett., 11 (2020), 7090-7095; arXiv:2006.01259 [physics.comp-ph], 2020.
I. J. Zucker, Madelung constants and lattice sums for invariant cubic lattice complexes and certain tetragonal structures, J. Phys. A: Math. Gen. 8 (11) (1975) 1734.
Wikipedia, Madelung constant

Cf. A085469, A088537, A090734.

digits = 105;
m0 = 50; (* initial number of terms *)
dm = 10; (* number of terms increment *)
dd = 10; (* precision excess *)
Clear[f];
f[n_, p_] := f[n, p] = (s = Sqrt[n^2 + p^2]; ((2 + (-1)^n) Csch[s*Pi])/s // N[#, digits + dd]&);
f[m_] := f[m] = Pi/2 - (7 Log[2])/2 + 4 Sum[f[n, p], {n, 1, m}, {p, 1, m}];
f[m = m0];
f[m += dm];
While[Abs[f[m] - f[m - dm]] > 10^(-digits - dd), Print["f(", m, ") = ", f[m]]; m += dm];
A185577 = f[m];
Clear[g];
g[m_] := g[m] = 12 Pi Sum[Sech[(Pi/2) Sqrt[(2 j + 1)^2 + (2 k + 1)^2]]^2, {j, 0, m}, {k, 0, m}] // N[#, digits + dd]&;
g[m = m0];
g[m += dm];
While[Abs[g[m] - g[m - dm]] > 10^(-digits - dd), Print["g(", m, ") = ", g[m]]; m += dm];
A085469 = g[m];
A181152 = Sqrt[3] (A085469 - 3 A185577)/4;
RealDigits[A181152, 10, digits][[1]] (* Jean-François Alcover, May 07 2021 *)

More terms (using the above comment from R. J. Mathar and terms from the b-files for A085469 and A185577) from Jon E. Schoenfield, Mar 10 2018
Definition corrected by Andrey Zabolotskiy, Oct 21 2019
a(88)-a(105) from Jean-François Alcover, May 07 2021

A093821 Decimal expansion of (2*(3 - sqrt(3)))/3.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Apr 16 2004

Area of lamina found by Pal in the Lebesgue minimal problem.

This appears also as the ratio r_3/r_4 of the outer radii of a regular triangle circumscribed by a regular quadrangle such that they share one vertex and the other two vertices of the triangle touch two sides of the quadrangle. The center of the quadrangle is displaced from the center of the triangle by d/r_4 = 1-2/sqrt(3) = 0.154700.. (see A020832). # R. J. Mathar, Jan 22 2013

			0.845299461...

Eric Weisstein's World of Mathematics, Lebesgue Minimal Problem

Cf. A010527, A093822.

evalf(2*(1-1/sqrt(3))) ; # R. J. Mathar, Jan 22 2013

A176053 Decimal expansion of (3+2*sqrt(3))/3.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 07 2010

Continued fraction expansion of (3+2*sqrt(3))/3 is A010696.
a(n) = A020832(n-1) for n > 1; a(1) = 2.
This equals the ratio of the radius of the outer Soddy circle and the common radius of the three kissing circles. See A343235, also for links. - Wolfdieter Lang, Apr 19 2021
Previous comment is, together with A246724, the answer to the 1st problem proposed during the 4th Canadian Mathematical Olympiad in 1972. - Bernard Schott, Mar 20 2022

			2.15470053837925152901...

Michael Doob, The Canadian Mathematical Olympiad & L'Olympiade Mathématique du Canada 1969-1993 - Canadian Mathematical Society & Société Mathématique du Canada, Problem 1, 1972, page 37, 1993.

Paolo Xausa, Table of n, a(n) for n = 1..10000
The IMO Compendium, Problem 1, 4th Canadian Mathematical Olympiad, 1972.
Michael Penn, An inscribed tower of squares, YouTube video, 2020.
Bernard Schott, Illustration of the Soddy circles.
Index to sequences related to Olympiads.

Cf. A002194 (sqrt(3)), A020832 (1/sqrt(75)), A010696 (repeat 2, 6).

Cf. A246724, A343235.

RealDigits[1+2/3Sqrt[3],10,100][[1]] (* Paolo Xausa, Aug 10 2023 *)

Equals 2 + A246724.

A041133 Denominators of continued fraction convergents to sqrt(75).

Original entry on oeis.org

1, 1, 2, 3, 50, 53, 103, 156, 2599, 2755, 5354, 8109, 135098, 143207, 278305, 421512, 7022497, 7444009, 14466506, 21910515, 365034746, 386945261, 751980007, 1138925268, 18974784295, 20113709563, 39088493858, 59202203421, 986323748594, 1045525952015

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 (0,0,0,52,0,0,0,-1).

Cf. A041132, A020832, A010153.

I:=[1, 1, 2, 3, 50, 53, 103, 156]; [n le 8 select I[n] else 52*Self(n-4)-Self(n-8): n in [1..40]]; // Vincenzo Librandi, Dec 11 2013

Denominator/@Convergents[Sqrt[75], 50] (* Vladimir Joseph Stephan Orlovsky, Jul 05 2011 *)
CoefficientList[Series[-(x^2 - x - 1) (x^4 + 3 x^2 + 1)/(x^8 - 52 x^4 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2013 *)
LinearRecurrence[{0,0,0,52,0,0,0,-1},{1,1,2,3,50,53,103,156},40] (* Harvey P. Dale, Aug 03 2024 *)

G.f.: -(x^2-x-1)*(x^4+3*x^2+1) / (x^8-52*x^4+1). - Colin Barker, Nov 13 2013

a(n) = 52*a(n-4) - a(n-8). - Vincenzo Librandi, Dec 11 2013

A270230 Decimal expansion of 3/(4*Pi).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Mar 13 2016

Consider generic prisms with triangular bases (tp), enclosed by a sphere, and let f(tp) be the fraction of the sphere volume occupied by any of them (i.e., the ratio of the prism volume to the sphere volume). Then this constant is the supremum of f(tp). It is attained by prisms which have as their base equilateral triangles with edge lengths r*sqrt(2), and rectangular side faces that are r*sqrt(2) wide and r*2/sqrt(3) high, where r is the radius of the enclosing, circumscribed sphere.
An intriguing fact is that the volume of such a best-fitting prism is exactly r^3. Hence, 1/a is the volume of a sphere with radius 1.
Examples of similar constants obtained for other shapes enclosed by spheres are: A020760 for cylinders and A165952 for cuboids.

			0.238732414637843003653325645058771543051689468610684673121501016...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Index entries for transcendental numbers

Cf. A002193, A019699 (one tenth of 1/a), A020760, A020832 (one tenth of 2/sqrt(3)), A165952.

First@ RealDigits[N[3/4/Pi, 120]] (* Michael De Vlieger, Mar 15 2016 *)

PARI
```
3/4/Pi
```

cp's OEIS Frontend

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