A336308 - OEIS

%I #30 Mar 30 2024 02:47:36

%S 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,

%T 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,

%U 6,3,1,3,4,8,2,2,2,0,1,5,6,0,3,5,6,3,0

%N Decimal expansion of (5/32)*Pi.

%C (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.

%H Robert Ferréol, <a href="https://mathcurve.com/courbes2d.gb/foliumsimple/foliumsimple.shtml">Simple folium</a>, Mathcurve.

%H <a href="/index/Cu#curves">Index to sequences related to curves</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F Equals Integral_{t=0..Pi} cos^6(t)/2 dt (area of simple folium).

%F From _Amiram Eldar_, Aug 13 2020: (Start)

%F Equals Integral_{x=0..oo} 1/(x^2 + 1)^4 dx.

%F Equals Integral_{x=-1..1} x^3 * arcsin(x) dx. (End)

%F 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

%e 0.4908738521234051935097880286374223256558077186523602...

%p evalf(5*Pi/32, 140);

%t RealDigits[5*Pi/32, 10, 100][[1]] (* _Amiram Eldar_, Jul 17 2020 *)

%o (PARI) 5*Pi/32 \\ _Michel Marcus_, Jul 17 2020

%Y 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).

%K nonn,cons

%O 0,1

%A _Bernard Schott_, Jul 17 2020