A336266 -id:A336266 - cp's OEIS frontend

A019694 Decimal expansion of 2*Pi/5.

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

Offset: 1

N. J. A. Sloane

Also, with proper offset, decimal expansion of the magnetic permeability of vacuum in SI units, mu_0 = 4*Pi*10^-7 N A^-2, an assigned metrological constant. [This exact expression for mu_0 was valid until the 2019 SI redefinition. In the New SI, mu_0 is numerically very close to that value but is determined only up to a certain error. - Andrey Zabolotskiy, May 22 2019]
Regarding these, see A003678 for general context notes, references and links. - Stanislav Sykora, Jun 16 2012
With offset 2 this is also the decimal expansion of 4*Pi, the surface area of a sphere whose diameter equals the square root of 4, hence its radius is 1. More generally x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Jan 18 2013, Oct 05 2013, Dec 22 2013
4*Pi is also the area of the domain bounded by the witch of Agnesi whose Cartesian equation is y = 8 / (x^2 + 4) and its asymptote. More generally (4*Pi) * a^2 is the area of the domain bounded by the witch of Agnesi whose Cartesian equation is y = (8*a^3) / (x^2 + 4*a^2) and its asymptote (Eric Weisstein's link, formula 6). - Bernard Schott, Jun 28 2023

			1.2566370614359172953850573533118....
mu_0 = 12.566370614359172953850573533118... 10^-7 N/A^2. - _Stanislav Sykora_, Jun 16 2012

Ivan Panchenko, Table of n, a(n) for n = 1..1000
NIST, magnetic constant mu_0.
Nobelprize.org, The Discovery of Quasicrystals (PDF).
Eric Weisstein's World of Mathematics, Witch of Agnesi.
Wikipedia, Crystallographic Restriction Theorem.
Index entries for sequences related to curves.
Index entries for transcendental numbers.

Other assigned constants: A003678, A072915, A081799, A182999, A213610, A213611, A213612, A213613, A213614.

Cf. A093828 (astroid), A180434 (loop of strophoid), A197723 (cardioid), A336266 (double egg), A336308 (ovoid).

Digits:=100: evalf(2*Pi/5); # Wesley Ivan Hurt, Jan 07 2017

RealDigits[ 2*Pi/5, 10, 111][[1]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 and modified by Robert G. Wilson v *)

PARI
```
2*Pi/5 \\ G. C. Greubel, Sep 11 2017
```

Equals Sum_{k>=1} sin(Pi*k/5)/k. - Amiram Eldar, Aug 12 2020

Equals -zeta(3/2)/(10*zeta(-1/2)). - Mats Granvik, May 28 2022

A057813 a(n) = (2n+1)(4n^2+4n+3)/3.

Original entry on oeis.org

1, 11, 45, 119, 249, 451, 741, 1135, 1649, 2299, 3101, 4071, 5225, 6579, 8149, 9951, 12001, 14315, 16909, 19799, 23001, 26531, 30405, 34639, 39249, 44251, 49661, 55495, 61769, 68499, 75701, 83391, 91585, 100299, 109549, 119351, 129721, 140675, 152229, 164399

Offset: 0

N. J. A. Sloane, Nov 07 2000

For n>0, 30*a(n) is the sum of the ten distinct products of 2*n-1, 2*n+1, and 2*n+3. For example, when n = 1, we sum the ten distinct products of 1, 3, and 5: 1*1*1 + 1*1*3 + 1*1*5 + 1*3*3 + 1*3*5 + 1*5*5 + 3*3*3 + 3*3*5 + 3*5*5 + 5*5*5 = 330 = 30*11 = 30*a(1). - J. M. Bergot, Apr 06 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (10).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

1/12*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.

Cf. A019560, A336266.

[(2*n+1)*(4*n^2+4*n+3)/3 : n in [0..50]] // Wesley Ivan Hurt, Apr 22 2014

A057813:=n->(2*n + 1)*(4*n^2 + 4*n + 3)/3; seq(A057813(n), n=0..50); # Wesley Ivan Hurt, Apr 06 2014

Table[(2*n + 1)*(4*n^2 + 4*n + 3)/3, {n, 0, 50}] (* David Nacin, Mar 01 2012 *)

P(x, y, z) = x^3 + x^2*y + x^2*z + x*y^2 + x*y*z + x*z^2 + y^3 + y^2*z + y*z^2 + z^3;
a(n) = P(2*n-1, 2*n+1, 2*n+3)/30; \\ Michel Marcus, Apr 22 2014

a(n) = 2*A050533(n) + 1. - N. J. A. Sloane, Sep 22 2004
G.f.: (1+7*x+7*x^2+x^3)/(1-x)^4. - Colin Barker, Mar 01 2012
G.f. for sequence with interpolated zeros: 1/(8*x)*sinh(8*arctanh(x)) = 1/(16*x)*( ((1 + x)/(1 - x))^4 - ((1 - x)/(1 + x))^4 ) = 1 + 11*x^2 + 45*x^4 + 119*x^6 + .... Cf. A019560. - Peter Bala, Apr 07 2017
E.g.f.: (3 + 30*x + 36*x^2 + 8*x^3)*exp(x)/3. - G. C. Greubel, Dec 01 2017
From Peter Bala, Mar 26 2024: (Start)
12*a(n) = (2*n + 1)*(a(n + 1) - a(n - 1)).
Sum_{n >= 0} (-1)^n/(a(n)*a(n+1)) = 3*Pi/16 - 1/2. Cf. A016754 and A336266. (End)

A336308 Decimal expansion of (5/32)*Pi.

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Jul 17 2020

(5*Pi/32)*a^2 is the area of a simple folium also called ovoid, or Kepler egg whose polar equation is r = a*cos^3(t) and Cartesian equation is (x^2+y^2)^2 = a * x^3. See the curve at the Mathcurve link.

			0.4908738521234051935097880286374223256558077186523602...

Robert Ferréol, Simple folium, Mathcurve.
Index to sequences related to curves.
Index entries for transcendental numbers

Cf. A019692 (2*Pi for deltoid), A122952 (3*Pi for cycloid and nephroid), A180434 (2-Pi/2 for Newton strophoid), A197723 (3*Pi/2 for cardioid), A336266 (3*Pi/16 for double egg).

Maple
```
evalf(5*Pi/32, 140);
```

RealDigits[5*Pi/32, 10, 100][[1]] (* Amiram Eldar, Jul 17 2020 *)

PARI
```
5*Pi/32 \\ Michel Marcus, Jul 17 2020
```

Equals Integral_{t=0..Pi} cos^6(t)/2 dt (area of simple folium).
From Amiram Eldar, Aug 13 2020: (Start)
Equals Integral_{x=0..oo} 1/(x^2 + 1)^4 dx.
Equals Integral_{x=-1..1} x^3 * arcsin(x) dx. (End)
Equals 5/9 - 10*Sum_{n >= 1} (-1)^(n+1)/(u(n)*u(-n)), where the polynomial u(n) = (2*n - 1)^2*(4*n^2 - 4*n + 9)/3 satisfies the difference equation 16*u(n) = (2*n - 1)*(u(n+1) - u(n-1)) and has its zeros on the vertical line Re(z) = 1/2 in the complex plane. Cf. A336266. - Peter Bala, Mar 25 2024

A376642 Decimal expansion of the area of Moss's egg constructed from a unit-hypotenuse right isosceles triangle.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Sep 30 2024

Moss's egg is an oval named by Dixon (1987) after Stephanie Moss. It is formed by four circular arcs. The shape is composed of the area of a half disk of radius 1/2, circular sector with radius 1-sqrt(2)/2 and central angle Pi/2, and two partially overlapping circular sectors with radius 1 and central angle Pi/4, whose common area is of the unit-hypotenuse right isosceles triangle.

The perimeter of the shape is (3-sqrt(2)/2)*Pi/2.

			0.99547375565275336709301228994445373849422162718740...

Robert Dixon, Mathographics, New York: Dover, 1987. See p. 5.
Anna Weltman, Not Your Average Maths Book, Wide Eyed Editions, 2022. See p. 43.

Michael Borcherds and Bill Lombard, Moss Egg by Robert Dixon, GeoGebra.
Eric Weisstein's World of Mathematics, Moss's Egg.
Wikipedia, Moss's egg.

Similar constants: A093731, A259830, A336266, A336308.

RealDigits[((3 - Sqrt[2])*Pi - 1)/4, 10, 120][[1]]

PARI
```
((3-quadgen(8))*Pi - 1)/4
```

Equals ((3-sqrt(2))*Pi - 1)/4.

cp's OEIS Frontend

A019694 Decimal expansion of 2*Pi/5.

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A057813 a(n) = (2*n+1)*(4*n^2+4*n+3)/3.

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A336308 Decimal expansion of (5/32)*Pi.

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Maple

Mathematica

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Formula

A376642 Decimal expansion of the area of Moss's egg constructed from a unit-hypotenuse right isosceles triangle.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A057813 a(n) = (2n+1)(4n^2+4n+3)/3.