A336266 - OEIS

%I #42 Mar 30 2024 02:42:43

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

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

%U 5,5,7,6,1,7,8,6,6,4,1,8,7,2,4,2,7,5,6,5

%N Decimal expansion of (3/16)*Pi.

%C Area of one egg of the "double egg" whose polar equation is r(t) = a * cos(t)^2 and a Cartesian equation is (x^2+y^2)^3 = a^2*x^4 is equal to (3/16)*Pi * a^2. See the curve at the Mathcurve link.

%D L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 586.

%H Peter Bala, <a href="/A336266/a336266.pdf">A note on A336266</a>

%H Robert Ferréol, <a href="https://mathcurve.com/courbes2d.gb/oeufdouble/oeufdouble.shtml">Double egg</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} (1/2) * cos(t)^4 * dt.

%F Equals Integral_{x=0..oo} 1/(x^2 + 1)^3 dx. - _Amiram Eldar_, Aug 13 2020

%F From _Peter Bala_, Mar 21 2024: (Start)

%F Equals 1/2 + Sum_{n >= 0} (-1)^n/(u(n)*u(-n)), where the polynomial u(n) = (2*n - 1)*(4*n^2 - 4*n + 3)/3 = A057813(n-1) has its zeros on the vertical line Re(z) = 1/2 in the complex plane. Cf. A336308.

%F Equals 1/2 + 1/(11 + 3/(12 + 15/(12 + 35/(12 + ... + (4*n^2 - 1)/(12 + ... ))))). See Lorentzen and Waadeland, p. 586, equation 4.7.10 with n = 2. (End)

%e 0.58904862254808623221174563436490679078696926...

%p evalf(3*Pi/16,140);

%t RealDigits[3*Pi/16, 10, 100][[1]] (* _Amiram Eldar_, Jul 15 2020 *)

%o (PARI) 3*Pi/16 \\ _Michel Marcus_, Jul 15 2020

%Y Cf. A259830 (length of an egg).

%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), A336308.

%K nonn,cons

%O 0,1

%A _Bernard Schott_, Jul 15 2020