Displaying 1-10 of 11 results found.
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
Numbers that are congruent to {2, 4} mod 6.
+10
43
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
COMMENTS
Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 19 ).
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
FORMULA
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 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 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)
MAPLE
seq(6*floor((n+1)/2) + 3 + (-1)^n, n=1..67); # Gary Detlefs, Mar 02 2010
MATHEMATICA
Flatten[Table[{6n - 4, 6n - 2}, {n, 40}]] (* Alonso del Arte, Oct 27 2014 *)
PROG
(Magma) [ n eq 1 select 2 else Self(n-1)+2*(1+n mod 2): n in [1..70] ]; // Klaus Brockhaus, Dec 13 2008 (PARI) first(n) = my(v = vector(n, i, 3*i - 1)); forstep(i = 2, n, 2, v[i]--); v \\ David A. Corneth, Oct 20 2017
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.
+10
12
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
COMMENTS
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
REFERENCES
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.
LINKS
The IMO Compendium, Problem 1, 4th Canadian Mathematical Olympiad, 1972.
FORMULA
Equals (2*sqrt(3) - 3)/3.
Equals -1 + sqrt(2) * sqrt(2-sqrt(2)) * sqrt(2-sqrt(2-sqrt(2))) * ... (Moreno and García, 2013). - Amiram Eldar, Jun 09 2022
EXAMPLE
0.154700538379251529018297561003914911295203502540253752...
MATHEMATICA
RealDigits[(2*Sqrt[3] - 3)/3, 10, 103] // First
Expansion of Jacobi theta function (theta_3(q^(1/3))-theta_2(q^3))/2/q^(1/12).
+10
7
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
FORMULA
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
EXAMPLE
1 + q^2 + q^4 + q^10 + q^14 + q^24 + q^30 + q^44 + q^52 + ...
MATHEMATICA
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 *)
PROG
(PARI) 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
Decimal expansion of 2*sqrt(3)/(3*Pi).
+10
7
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
COMMENTS
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.
LINKS
Eric Weisstein's World of Mathematics, Cube.
EXAMPLE
0.3675525969478613663408843322086462942649243202444271018662440135273535356...
MATHEMATICA
RealDigits[(2*Sqrt[3])/(3Pi), 10, 120][[1]] (* Harvey P. Dale, Oct 08 2012 *)
Decimal expansion of Madelung constant (negated) for the CsCl structure.
+10
5
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
COMMENTS
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
MATHEMATICA
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];
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];
Decimal expansion of (2*(3 - sqrt(3)))/3.
+10
4
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
COMMENTS
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
Decimal expansion of (3+2*sqrt(3))/3.
+10
4
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
COMMENTS
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
REFERENCES
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.
LINKS
The IMO Compendium, Problem 1, 4th Canadian Mathematical Olympiad, 1972.
EXAMPLE
2.15470053837925152901...
MATHEMATICA
RealDigits[1+2/3Sqrt[3], 10, 100][[1]] (* Paolo Xausa, Aug 10 2023 *)
Denominators of continued fraction convergents to sqrt(75).
+10
2
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
FORMULA
G.f.: -(x^2-x-1)*(x^4+3*x^2+1) / (x^8-52*x^4+1). - Colin Barker, Nov 13 2013
MATHEMATICA
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 *)
PROG
(Magma) 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
Decimal expansion of 3/(4*Pi).
+10
1
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
COMMENTS
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.
EXAMPLE
0.238732414637843003653325645058771543051689468610684673121501016...
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