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

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

Author

Bernard Schott, Jul 17 2020

Keywords

• nonn
• cons

Comments

(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.

Examples

			0.4908738521234051935097880286374223256558077186523602...
		

Links

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

Crossrefs

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).

Programs

Maple

evalf(5*Pi/32, 140);

Mathematica

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

PARI

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

Formula

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